Open AccessArticle

Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions

Constantin Fetecău

¹ and

Costică Moroşanu

^2,*

Academy of Romanian Scientists, 54 Splaiul Independentei, 050094 Bucharest, Romania

Department of Mathematics, “Alexandru Ioan Cuza” University, Bd. Carol I, 11, 700506 Iaşi, Romania

Author to whom correspondence should be addressed.

Axioms 2023, 12(4), 406; https://doi.org/10.3390/axioms12040406

Submission received: 26 February 2023 / Revised: 7 April 2023 / Accepted: 19 April 2023 / Published: 21 April 2023

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

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Abstract

Two main topics are addressed in the present paper, first, a rigorous qualitative study of a second-order reaction–diffusion problem with non-linear diffusion and cubic-type reactions, as well as inhomogeneous dynamic boundary conditions. Under certain assumptions about the input data:

g_{_{d}} (t, x)

g_{_{f r}} (t, x)

U_{0} (x)

and

ζ_{0} (x)

, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a solution in the space

W_{p}^{1, 2} (Q) \times W_{p}^{1, 2} (Σ)

. Here, we extend previous results, enabling new mathematical models to be more suitable to describe the complexity of a wide class of different physical phenomena of life sciences, including moving interface problems, material sciences, digital image processing, automatic vehicle detection and tracking, the spread of an epidemic infection, semantic image segmentation including U-Net neural networks, etc. The second goal is to develop an iterative splitting scheme, corresponding to the non-linear second-order reaction–diffusion problem. Results relating to the convergence of the approximation scheme and error estimation are also established. On the basis of the proposed numerical scheme, we formulate the algorithm alg-frac_sec-ord_dbc, which represents a delicate challenge for our future works. The benefit of such a method could simplify the process of numerical computation.

Keywords:

boundary value problems for non-linear parabolic PDE; fractional step method; convergence of numerical methods; numerical algorithm; error analysis; dynamic boundary conditions

MSC:

35K55; 65N06; 65N12; 65YXX; 80AXX

1. Introduction

Considering the following non-linear second-order reaction–diffusion problem:

\{\begin{matrix} p_{_{1}} \frac{\partial}{\partial t} U (t, x) - p_{_{2}} div (K (t, x, U (t, x)) \nabla U (t, x)) \\ = p_{_{r}} [U (t, x) - U^{3} (t, x)] + p_{_{s}} g_{_{d}} (t, x) & in Q \\ p_{_{2}} \frac{\partial}{\partial n} U + p_{_{1}} \frac{\partial}{\partial t} U - Δ_{_{Γ}} U + p_{_{t}} U = g_{_{f r}} (t, x) & on Σ \\ U (0, x) = U_{0} (x) & on Ω, \end{matrix}

(1)

where

Ω \subset {I R}^{n}

n \leq 3

is a compact domain with a

C^{2}

boundary

\partial Ω = Γ

[0, T]

a generic time interval,

Q = (0, T] \times Ω

Σ = (0, T] \times \partial Ω

and:

$t \in (0, T]$ , $x = (x_{1}, \dots, x_{n}) \in Ω$ ;
$p_{_{1}}$ , $p_{_{2}}$ , $p_{_{r}}$ , $p_{_{s}}$ and $p_{_{t}}$ are positive parameters;
$\frac{\partial}{\partial s} U (s, \cdot)$ ( $U_{s}$ in short) is the partial derivative of $U (s, \cdot)$ (U in short) relative to $s \in (0, T]$ ;
$U (s, y)$ , $(s, y) \in Q$ , is the unknown function (the order parameter in Q, for example). $\nabla U (s, y) = U_{y} (s, y)$ ( $\nabla U = U_{y}$ ) denotes the gradient of $U (s, y)$ in $y, y \in Ω$ (see [1,2,3] for more details);
$K (s, y, U (s, y))$ is the mobility (attached to the solution $U (s, y)$ , $(s, y) \in Q$ , to Equation (1)) (see [2,3,4] for more details);
$g_{_{d}} (s, y) \in L^{p} (Q)$ is the distributed control (see Remark 1 below), where

$p \geq 2;$

(2)
$g_{_{f r}} (s, y) \in W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)$ is the boundary control (see Remark 1 below);
$U_{0} \in W_{\infty}^{2 - \frac{2}{p}} (Ω)$ verifying

$p_{_{2}} \frac{\partial}{\partial n} U_{0} - Δ_{_{Γ}} U_{0} + p_{_{t}} U_{0} = g_{_{f r}} (0, x);$
n = $n (x)$ has the same meaning as in [5];
$Δ_{_{Γ}}$ has the same meaning as in [6];

Remark 1.

The given functions

g_{_{d}}

and

g_{_{f r}}

in (1), can be interpreted as distributed and boundary control, respectively, opening a large field of applications for the non-linear second-order problem (1), such as optimal control.

For convenience, let us write (1) in the following form

\{\begin{matrix} p_{_{1}} \frac{\partial}{\partial t} U (t, x) - p_{_{2}} \frac{\partial}{\partial U_{x_{j}}} [K (t, x, U (t, x)) U_{x_{i}}] U_{x_{j} x_{i}} \\ = A (t, x, U (t, x), U_{x_{i}} (t, x)) + p_{_{r}} [U (t, x) - U^{3} (t, x)] + p_{_{s}} g_{_{d}} (t, x) & in Q \\ p_{_{2}} \frac{\partial}{\partial n} U + p_{_{1}} \frac{\partial}{\partial t} U - Δ_{_{Γ}} U + p_{_{t}} U = g_{_{f r}} (t, x) & on Σ \\ U (0, x) = U_{0} (x) & on Ω, \end{matrix}

(3)

where

U_{x_{j} x_{i}} = \frac{\partial^{2}}{\partial x_{j} \partial x_{i}} U (t, x), i, j = 1, \dots, n

, and

A (t, x, U (t, x), U_{x_{i}} (t, x)) = \frac{\partial}{\partial U} (K (t, x, U) U_{x_{i}}) U_{x_{i}} + \frac{\partial}{\partial x_{i}} (K (t, x, U) U_{x_{i}}), i = 1, \dots, n .

(4)

As in [1,2,3,5,6,7,8,9], we recall that Equation (1)

_{1}

is a quasi-linear one, i.e.,

a_{i} (t, x, U (t, x), U_{x} (t, x)) = K (t, x, U (t, x)) U_{x_{i}} (t, x), i = 1, \dots, n

and

a (t, x, U (t, x), U_{x} (t, x)) = - p_{_{r}} [U (t, x) - U^{3} (t, x)] - p_{_{s}} g_{_{d}} (t, x) .

On the other hand, the problem in (3)

_{1}

is similar to in [10] (p. 3, relation (2.4)), where, for

i = 1, \dots, n,

a_{i j} (t, x, U (t, x), U_{x} (t, x)) = \frac{\partial}{\partial U_{_{x_{j}}}} a_{i} (t, x, U (t, x), U_{x} (t, x)) = \frac{\partial}{\partial U_{_{x_{j}}}} [K (t, x, U (t, x)) U_{x_{i}} (t, x)],

and

a (t, x, U (t, x), U_{x} (t, x)) = - A (t, x, U (t, x), U_{x} (t, x)) - p_{_{r}} [U (t, x) - U^{3} (t, x)] - p_{_{s}} g_{_{d}} (t, x),

while (3)

_{2}

are of the second type, namely

\frac{\partial}{\partial n} U (t, x) = a_{i j} (t, x, U (t, x), U_{x} (t, x)) U_{x_{j}} (t, x) c o s α_{i},

and

{ψ (t, x, U) |}_{_{Σ}} = p_{_{1}} \frac{\partial}{\partial t} U - Δ_{_{Γ}} U + p_{_{t}} U - g_{_{f r}} (t, x)

(5)

(see [10] (p. 475, relation (7.2))).

Moreover, we consider that Equations (1)

_{1}

and (3)

_{1}

are uniformly parabolic, i.e.,

ν_{_{1}} (| U |) ζ^{2} \leq \frac{\partial}{\partial z_{j}} a_{i} (s, y, U (s, y), z (s, y)) ζ_{i} ζ_{j} \leq ν_{_{2}} (| U |) ζ^{2}

(6)

for arbitrary

U (s, y)

and

z (s, y)

(s, y) \in Q

, and

ζ = (ζ_{1}, \dots, ζ_{n})

for an arbitrary real vector (see [5] for more details).

Equation (1)

_{1}

was initially introduced by Allen and Cahn (see [5,11] and references therein) to describe the motion of anti-phase boundaries in crystalline solids. In fact, the Allen–Cahn model is widely applied to moving interface problems, such as the mixture of two incompressible fluids, the nucleation of solids, vesicle membranes, etc. Furthermore, the non-linear parabolic Equation (1)

_{1}

appears in the Caginalp’s phase-field transition system (see [2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,20,21,22]), describing the transition between phases (solid and liquid) (see [17], for example).

In the present paper we investigate the solvability of boundary value problems of the form (1) or (3) in the class

W_{p}^{1, 2} (Q)

. The new model expressed in (1) stands out by the presence of parameters

p_{_{1}}

p_{_{2}}

p_{_{r}}

p_{_{s}}

p_{_{t}}

K (s, y, U (s, y)),

and

(s, y) \in Q

, the principal part being in the divergence form and by considering a non-linear reaction term (see [5,11] and references therein). The most important aspect in our paper concerns inhomogeneous dynamic boundary conditions. Thus, we more precisely define the significant aspects of the physical features. In this regard, we advise applying (1) or (3), to the moving interface problems (see [5,7,8,11,12,13,14,15]), anisotropy effects (see [3,4,5,6,9,11,16,17,18,19,20,21,22]), image de-noising and segmentation (see [2,4] and references therein), etc. Let us point out that the following assumption is satisfied (see [20]):

H_{0} : (U - U^{3}) {| U |}^{3 p - 4} U \leq 1 + {| U |}^{3 p - 1} - {| U |}^{3 p} .

2. Results—Theorem 1

In order to approach the problem in (3) (or (1)), we use the same ideas as in [1,6,7,9]. In this respect we introduce a new variable

ζ (t, x) = U (t, x)

ζ (0, x) = U_{0} (x)

Γ

(see [10] (6.2)). Correspondingly, (3)

_{2}

is approached in the following

\{\begin{matrix} U (t, x) = ζ (t, x) & on Σ \\ p_{_{2}} \frac{\partial}{\partial n} U + p_{_{1}} \frac{\partial}{\partial t} ζ (t, x) - Δ_{_{Γ}} ζ (t, x) + p_{_{t}} ζ (t, x) = g_{_{f r}} (t, x) & on Σ \\ ζ (0, x) = ζ_{0} (x) & x \in Γ . \end{matrix}

(7)

Accordingly, the non-linear second-order boundary value problem (3) can be written suitably as follows

\{\begin{matrix} p_{_{1}} \frac{\partial}{\partial t} U (t, x) - p_{_{2}} \frac{\partial}{\partial U_{x_{j}}} [K (t, x, U (t, x)) U_{x_{i}} (t, x)] U_{x_{j} x_{i}} \\ = A (t, x, U (t, x), U_{x_{i}} (t, x)) + p_{_{r}} [U (t, x) - U^{3} (t, x)] + p_{_{s}} g_{_{d}} (t, x) & in Q \\ U (t, x) = ζ (t, x) & on Σ \\ p_{_{2}} \frac{\partial}{\partial n} U + p_{_{1}} \frac{\partial}{\partial t} ζ - Δ_{_{Γ}} ζ + p_{_{t}} ζ = g_{_{f r}} (t, x) & on Σ \\ U (0, x) = U_{0} (x) & on Ω \\ ζ (0, x) = ζ_{0} (x) & x \in Γ, \end{matrix}

(8)

where

A (t, x, U (t, x), U_{x_{i}} (t, x))

is defined by (4),

U_{0} (x) = ζ_{0} (x)

Γ

and

ζ_{0} (x) \in W_{\infty}^{2 - \frac{2}{p}} (Γ)

Definition 1.

Any solution

(U (t, x), ζ (t, x))

to problem (8) is called the classical solution if it is continuous in

\bar{Q}

, with continuous derivatives

U_{t}

U_{x}

and

U_{x x}

in Q and

ζ_{t}

ζ_{x}

, and

ζ_{x x}

in Σ, satisfying Equation (8)

_{1}

at all points

(t, x) \in Q

and satisfying conditions (8)

_{2 - 3}

and (8)

_{4 - 5}

on the lateral surface Σ of the cylinder Q for

t = 0

, respectively.

Our main results regarding the existence, uniqueness and regularity of solutions to problem (8) (the well-posedness of the solutions to the non-linear second-order boundary value problems (1) or (3)) are presented below.

Theorem 1.

Suppose

(U (t, x), ζ (t, x)) \in C^{1, 2} (Q) \times C^{1, 2} (Σ)

is a classical solution to problem (8), and for positive numbers M,

M_{0}

m_{1}

M_{1}

M_{2}

M_{3}

M_{4}

and

M_{5}

one has

I₁.: $| U (t, x) | < M$ for any $(t, x) \in Q$ and for any $z (t, x)$ , the map $K (t, x, z)$ is continuous, differentiable in x, where its x-derivatives are bounded, satisfy (6), and

$0 < K_{m i n} \leq K (t, x, U (t, x)) < K_{m a x}, f o r (t, x) \in Q,$

(9)

$\begin{matrix} \sum_{i = 1}^{n} [| a_{i} (t, x, U (t, x), z (t, x)) | + |\frac{\partial}{\partial U} a_{i} (t, x, U (t, x), z (t, x))|] (1 + | z |) \\ + \sum_{i, j = 1}^{n} |\frac{\partial}{\partial x_{j}} a_{i} (t, x, U (t, x), z (t, x))| + | U (t, x) | \leq M_{0} {(1 + | z |)}^{2} . \end{matrix}$

(10)
I₂.: For any sufficiently small $ε > 0$ , the functions $U (t, x)$ and $K (t, x, U (t, x))$ satisfy the relations
${∥ U ∥}_{_{L^{s} (Q)}} \leq M_{_{2}}$ , $∥ K (t, x, U (t, x)) U_{x_{i}} ∥_{_{L^{r} (Q)}} < M_{_{3}}, i = 1, \dots, n,$
where

$\begin{matrix} r = \{\begin{matrix} max {p, 4} & p \neq 4 \\ 4 + ε & p = 4, \end{matrix} & s = \{\begin{matrix} max {p, 2} & p \neq 2 \\ 2 + ε & p = 2 . \end{matrix} \end{matrix}$

Then, when

\forall g_{_{d}} \in L^{p} (Q)

U_{0} \in W_{\infty}^{2 - \frac{2}{p}} (Ω)

ζ_{0} (x) \in W_{\infty}^{2 - \frac{2}{p}} (Γ)

g_{_{f r}} \in W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)

, with

p \neq \frac{3}{2}

, there exists a unique solution

(U, ζ) \in W_{p}^{1, 2} (Q) \times W_{p}^{1, 2} (Σ)

to (8) which satisfies

\begin{matrix} {∥ U ∥}_{W_{p}^{1, 2} (Q)} + {∥ ζ ∥}_{W_{p}^{1, 2} (Σ)} \\ \leq C {1 + ∥ U_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Ω)} + ∥ ζ_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Γ)} + ∥ U_{0} ∥_{L^{3 p - 2} (Ω)}^{\frac{3 p - 2}{p}} + {∥ ζ_{0} ∥}_{L^{3 p - 2} (Γ)}^{\frac{3 p - 2}{p}} \\ + ∥ g_{_{d}} ∥_{L^{3 p - 2} (Q)}^{\frac{3 p - 2}{p}} + ∥ g_{_{_{f} r}} ∥_{L^{3 p - 2} (Σ)}^{\frac{3 p - 2}{p}} + {∥ g_{_{f r}} ∥}_{W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)}\}, \end{matrix}

(11)

where

C > 0

does not depend on U, ζ,

g_{_{d}}

, or

g_{_{f r}}

(U^{1}, ζ^{1})

and

(U^{2}, ζ^{2})

are solutions to (8) which correspond to

(U_{0}^{1}, ζ_{0}^{1})

(U_{0}^{2}, ζ_{0}^{2})

\in W_{\infty}^{2 - \frac{2}{p}} (Ω) \times W_{\infty}^{2 - \frac{2}{p}} (Γ)

g_{_{d}}^{1}

g_{_{d}}^{2}

g_{_{f r}}^{1}

and

g_{_{f r}}^{2}

, respectively, then

∥ U^{1} ∥_{W_{p}^{1, 2} (Q)}, {∥ U^{2} ∥}_{W_{p}^{1, 2} (Q)} \leq M_{4},

(12)

∥ ζ^{1} ∥_{W_{p}^{1, 2} (Σ)}, {∥ ζ^{2} ∥}_{W_{p}^{1, 2} (Σ)} \leq M_{5},

(13)

and the following holds

\begin{matrix} \underset{(t, x) \in Q}{m a x} | U^{1} - U^{2} | & + \underset{(t, x) \in Σ}{m a x} | ζ^{1} - ζ^{2} | \\ \leq C_{1} e^{C T} m a x \{\underset{(t, x) \in Ω}{m a x} | U_{0}^{1} - U_{0}^{2} |, \underset{(t, x) \in Γ}{m a x} | ζ_{0}^{1} - ζ_{0}^{2} |, \\ \underset{(t, x) \in Q}{m a x} | g_{_{d}}^{1} - g_{_{d}}^{2} |, \underset{(t, x) \in Σ}{m a x} | g_{_{f r}}^{1} - g_{_{f r}}^{2} |\}, \end{matrix}

(14)

where

C_{1} > 0

and

C > 0

, do not depend on

{U^{1}, ζ^{1}, g_{_{d}}^{1}

g_{_{f r}}^{1}, U_{0}^{1}, ζ_{0}^{1}}

and

\{U^{2}, ζ^{2}, g_{_{d}}^{2}, g_{_{f r}}^{2}, U_{0}^{2}, ζ_{0}^{2}\}

. In particular, the uniqueness of the solution to (8) holds.

As far as the techniques used in this paper are concerned, it should be noted that we derive the a priori estimates for

L^{p} (Q)

and

L^{p} (Σ)

. Moreover, basic tools in our approach are:

the Leray–Schauder degree theory (see [15] (p. 221) and reference therein);
the $L^{p}$ theory of linear and quasi-linear parabolic equations [10];
Green’s first identity

$\begin{matrix} - \int_{Ω} y div z d x = \int_{Ω} \nabla y \cdot z d x - \int_{\partial Ω} y \frac{\partial}{\partial n} z d γ, \\ - \int_{Ω} y Δ z d x = \int_{Ω} \nabla y \cdot \nabla z d x - \int_{\partial Ω} y \frac{\partial}{\partial n} z d γ, \end{matrix}$

(15)

for any scalar-valued function y and z in a continuously differentiable vector field in n dimensional space;
the Lions and Peetre embedding theorem [1] (p. 100) to ensure the existence of a continuous embedding $W_{p}^{1, 2} (Q) \subset L^{μ} (Q)$ , where the number $μ$ is defined as follows (see (2))

$μ = \{\begin{matrix} any positive number \geq 3 p & if \frac{1}{p} - \frac{2}{n + 2} \leq 0, \\ \frac{p (n + 2)}{n + 2 - 2 p} & if \frac{1}{p} - \frac{2}{n + 2} > 0 . \end{matrix}$

(16)

For a given positive integer k and

1 \leq p \leq \infty

, we denote by

W_{p}^{k, 2 k} (Q)

the Sobolev space on Q:

W_{p}^{k, 2 k} (Q) = \{y \in L^{p} (Q) : \frac{\partial^{i}}{\partial t^{i}} \frac{\partial^{j}}{\partial x^{j}} y \in L^{p} (Q), for 2 i + j \leq 2 k\},

i.e., the spaces of functions whose t- and x-derivatives up to the order k and

2 k

, respectively, belong to

L^{p} (Q)

. Furthermore, we use the Sobolev spaces

W_{p}^{i} (Ω)

and

W_{p}^{\frac{i}{2}, i} (Σ)

with the non-integral i for the initial and boundary conditions, respectively, (see [10] (p. 70 and 81)).

Furthermore, we use the set

C^{1, 2} (\bar{D})

(

C^{1, 2} (D)

) of all continuous functions in

\bar{D}

(in D) with continuous derivatives

u_{t}

u_{x}

, and

u_{x x}

\bar{D}

(in D) (

D = Q

D = Σ

), as well as the Sobolev spaces

W_{p}^{ℓ} (Ω)

, and

W_{p}^{ℓ, ℓ / 2} (Σ)

with non-integral ℓ for the initial and boundary conditions, respectively (see [10] (p. 8, p. 70 and p. 81)).

In the following we will denote by C some positive constants.

3. Proof of the Main Result—Theorem 1

We consider

B = W_{p}^{0, 1} (Q) \cap L^{3 p} (Q) \times L^{p} (Σ)

as a suitable Banach space, with the norm

{∥ \cdot ∥}_{B}

expressed by

∥ (φ, \bar{φ}) ∥_{B} = {∥ φ ∥}_{L^{p} (Q)} + ∥ φ_{x} ∥_{L^{p} (Q)} + {∥ \bar{φ} ∥}_{L^{p} (Σ)},

and a non-linear operator

H : B \times [0, 1] \to B

defined by

(U, ζ) = H (φ, \bar{φ}, λ) = (U (φ, \bar{φ}, λ), ζ (φ, \bar{φ}, λ)) \forall (φ, \bar{φ}) \in B, \forall λ \in [0, 1],

(17)

where

(U (φ, \bar{φ}, λ), ζ (φ, \bar{φ}, λ)

is a unique solution to the following linear second-order boundary value problem

\{\begin{matrix} p_{_{1}} \frac{\partial}{\partial t} U - p_{_{2}} [λ \frac{\partial}{\partial φ_{x_{j}}} (K (t, x, φ) φ_{x_{i}}) - (1 - λ) δ_{i}^{j}] U_{x_{i} x_{j}} \\ = λ \{A (t, x, φ, φ_{x_{i}}) + p_{_{r}} [φ (t, x) - φ^{3} (t, x)] + p_{_{s}} g_{_{d}} (t, x)\} & in Q \\ U (t, x) = ζ (t, x) & on Σ \\ U (0, x) = λ U_{0} (x) & on Ω \\ p_{_{2}} \frac{\partial}{\partial n} U + p_{_{1}} \frac{\partial}{\partial t} ζ - Δ_{_{Γ}} ζ + p_{_{t}} ζ = λ g_{_{f r}} (t, x) & on Σ \\ ζ (0, x) = λ ζ_{0} (x) & x \in Γ . \end{matrix}

(18)

Remark 2.

The non-linear operator H in (17) depends on

λ \in [0, 1]

and its fixed point for

λ = 1

is a solution to problem (18).

Proof.

We now prove that the non-linear operator H, defined in (17), is well-defined, continuous and compact.

From the right-hand side of (17)

_{1}

, it follows that,

\forall (φ, \bar{φ}) \in B

, then

φ^{3} \in L^{p} (Q)

and thus

A (t, x, φ, φ_{x_{i}}) + p_{_{r}} [φ (t, x) - φ^{3} (t, x)] + p_{_{s}} g_{_{d}} (t, x) \in L^{p} (Q)

. Using the

L^{p}

theory of linear parabolic equations (see [10]), the solution

(U, ζ)

to problem (18) exists and it is unique with

(U, ζ) = (U (φ, \bar{φ}, λ), ζ (φ, \bar{φ}, λ)) \in B, \forall (φ, \bar{φ}) \in B, \forall λ \in [0, 1] .

(19)

Using the continuous inclusions (see [6])

\{\begin{matrix} W_{p}^{1, 2} (Q) \subset B \subset L^{p} (Q) \\ W_{p}^{1, 2} (Σ) \subset L^{p} (Σ), \end{matrix}

(20)

we obtain

H (φ, \bar{φ}, λ) = (U, ζ) \in B

for all

(φ, \bar{φ}) \in B

and

\forall λ \in [0, 1]

, meaning the non-linear operator H is well defined.

Now, using the ideas from [1,2,3,4,5,6,7,9,16,20], let

φ^{n} \to φ

W_{p}^{0, 1} (Q) \cap L^{3 p} (Q)

{\bar{φ}}^{n} \to \bar{φ}

L^{p} (Σ)

and

λ^{n} \to λ

[0, 1]

. Using the notations

\begin{array}{l} (U^{n, λ_{n}}, ζ^{n, λ_{n}}) = H (φ^{n}, {\bar{φ}}^{n}, λ^{n}) \\ (U^{n, λ}, ζ^{n, λ}) = H (φ^{n}, {\bar{φ}}^{n}, λ), \\ (U^{λ}, ζ^{λ}) = H (φ, \bar{φ}, λ), \end{array}

we obtain

∥ u^{n, λ_{n}} - u^{n, λ} ∥_{W_{p}^{1, 2} (Q)} + {∥ ζ^{n, λ_{n}} - ζ^{n, λ} ∥}_{W_{p}^{1, 2} (Σ)} \to 0 for n \to \infty

(21)

and

∥ u^{n, λ} - u^{λ} ∥_{W_{p}^{1, 2} (Q)} + {∥ ζ^{n, λ} - ζ^{λ} ∥}_{W_{p}^{1, 2} (Σ)} \to 0 for n \to \infty .

(22)

The continuous embedding of (20), (21), and (22) allows us to derive the continuity of the non-linear operator H, introduced in (17). Furthermore, H is compact, easily written as

B \times [0, 1] \to W_{p}^{1, 2} (Q) \times W_{p}^{1, 2} (Σ) ↪ B = W_{p}^{0, 1} (Q) \cap L^{3 p} (Q) \times L^{p} (Σ),

where the second map is a compact inclusion (see [1] (p. 100)).

Next, we look at a positive number R, such that (see (17))

(U, ζ, λ) \in B \times [0, 1] with (U, ζ) = H (U, ζ, λ) ⟹ {∥ (U, ζ) ∥}_{B} < R .

(23)

The above expression

(U, ζ) = H (U, ζ, λ)

can be written as (see (1), (8) and (18))

\{\begin{matrix} p_{_{1}} \frac{\partial}{\partial t} U - λ p_{_{2}} div (K (t, x, U) \nabla U) - (1 - λ) p_{_{2}} Δ U \\ = λ [p_{_{r}} [U (t, x) - U^{3} (t, x)] + p_{_{s}} g_{_{d}} (t, x)] & in Q \\ U (t, x) = ζ (t, x) & on Σ \\ U (0, x) = λ U_{0} (x) & on Ω \\ p_{_{2}} \frac{\partial}{\partial n} U + p_{_{1}} \frac{\partial}{\partial t} ζ - Δ_{_{Γ}} ζ + p_{_{t}} ζ = λ g_{_{f r}} (t, x)] & on Σ \\ ζ (0, x) = λ ζ_{0} (x) & x \in Γ . \end{matrix}

(24)

Multiplying (24)

_{1}

{| U |}^{3 p - 4} U

and integrating over

Q_{s} : = (0, s) \times Ω

s \in (0, T]

, we obtain

\begin{matrix} \frac{p_{_{1}}}{3 p - 2} \int_{Ω} {| U (s, x) |}^{3 p - 2} d x \\ - λ p_{_{2}} \int_{Q_{s}} div (K (τ, x, U) \nabla U) {| U |}^{3 p - 4} U d τ d x \\ - (1 - λ) p_{_{2}} \int_{Q_{s}} Δ U {| U |}^{3 p - 4} U d τ d x \\ = λ p_{_{r}} \int_{Q_{s}} [U (τ, x) - U^{3} (τ, x)] {| U |}^{3 p - 4} U d τ d x + λ p_{_{s}} \int_{Q_{s}} g_{_{d}} (τ, x) {| U |}^{3 p - 4} U d τ d x . \end{matrix}

(25)

To process the terms

\int_{Q_{s}} div (K (τ, x, U) \nabla U) {| U |}^{3 p - 4} U d τ d x

and

\int_{Q_{s}} Δ U {| U |}^{3 p - 4} U d τ d x, in (25)

we use Green’s first identity (15)

_{1}

and (15)

_{2}

, respectively, to obtain

\begin{matrix} - λ p_{_{2}} \int_{Q_{s}} div (K (τ, x, U) \nabla U) {| U |}^{3 p - 4} U d τ d x \\ = λ p_{_{2}} \int_{Q_{s}} K (τ, x, U) \nabla U \cdot \nabla ({| U |}^{3 p - 4} U) d τ d x + λ \int_{Σ_{s}} {| U |}^{3 p - 4} U (- p_{_{2}} \frac{\partial}{\partial n} U) d τ d γ, \end{matrix}

(26)

\begin{matrix} - (1 - λ) p_{_{2}} \int_{Q_{s}} Δ U {| U |}^{3 p - 4} U d τ d x \\ = (1 - λ) 3 (p - 1) p_{_{2}} \int_{Q_{s}} {| \nabla U |}^{2} {| U |}^{3 p - 4} d τ d x + (1 - λ) \int_{Σ_{s}} {| U |}^{3 p - 4} U (- p_{_{2}} \frac{\partial}{\partial n} U) d τ d γ, \end{matrix}

(27)

where

Σ_{s} = (0, s) \times \partial Ω

s \in (0, T]

and

- p_{_{2}} \frac{\partial}{\partial n} U = p_{_{1}} \frac{\partial}{\partial t} ζ - Δ_{_{Γ}} ζ + p_{_{t}} ζ - λ g_{_{f r}}

(see (24)

_{4}

Combining the above equality with the boundary condition in (24)

_{2}

, the left inequality in (9), and the relations (26), (27), and (25) leads us to the following inequality

\begin{matrix} \frac{p_{_{1}}}{3 p - 2} \int_{Ω} {| U (s, x) |}^{3 p - 2} d x + λ \frac{p_{_{1}}}{3 p - 2} \int_{Γ} {| ζ (s, x) |}^{3 p - 2} d γ + (1 - λ) \frac{p_{_{1}}}{3 p - 2} \int_{Γ} {| ζ (s, x) |}^{3 p - 2} d γ \\ + λ p_{_{2}} \int_{Q_{s}} K (τ, x, U) \nabla U \cdot \nabla ({| U |}^{3 p - 4} U) d τ d x + (1 - λ) 3 (p - 1) p_{_{2}} \int_{Q_{s}} {| \nabla U |}^{2} {| U |}^{3 p - 4} d τ d x \\ + λ p_{_{t}} \int_{Σ_{s}} {| ζ (τ, x) |}^{3 p - 2} d τ d γ + (1 - λ) p_{_{t}} \int_{Σ_{s}} {| ζ (τ, x) |}^{3 p - 2} d τ d γ \\ + λ \int_{Σ_{s}} \nabla_{_{Γ}} ({| ζ |}^{3 p - 3}) \cdot \nabla_{_{Γ}} ζ d τ d γ + (1 - λ) \int_{Σ_{s}} \nabla_{_{Γ}} ({| ζ |}^{3 p - 3}) \cdot \nabla_{_{Γ}} ζ d τ d γ \\ \leq λ \frac{p_{_{1}}}{3 p - 2} \int_{Ω} | U_{0} {(x) |}^{3 p - 2} d x + \frac{p_{_{1}}}{3 p - 2} \int_{Γ} {| ζ_{0} (x) |}^{3 p - 2} d γ \\ + λ p_{_{r}} \int_{Q_{s}} [U (τ, x) - U^{3} (τ, x)] {| U |}^{3 p - 4} U d τ d x \\ + λ p_{_{s}} \int_{Q_{s}} g_{_{d}} {(τ, x) | U |}^{3 p - 4} U d τ d x + λ \int_{Σ_{t}} g_{_{_{f} r}} (τ, x) {| U |}^{3 p - 4} U d τ d γ \end{matrix}

(28)

for all

s \in (0, T]

. The last two terms in the above inequalities can be manipulated via Hölder and Cauchy’s inequality giving us the following estimates

\begin{matrix} a . λ p_{_{s}} \int_{Q_{s}} g_{_{d}} (τ, x) {| U |}^{3 p - 4} U d τ d x \\ \leq \frac{(3 p - 2) - 1}{3 p - 2} ε^{\frac{3 p - 2}{3 p - 3}} \int_{Q_{s}} {| U |}^{3 p - 2} d τ d x + λ p_{_{s}} \frac{1}{3 p - 2} ε^{- (3 p - 2)} \int_{Q_{s}} {| g_{_{d}} |}^{3 p - 2} d τ d x, \\ b . λ \int_{Σ_{s}} g_{_{_{f} r}} (τ, x) {| U |}^{3 p - 4} U d τ d γ \\ \leq \frac{(3 p - 2) - 1}{3 p - 2} ε^{\frac{3 p - 2}{3 p - 3}} \int_{Σ_{s}} {| U |}^{3 p - 2} d τ d γ + λ \frac{1}{3 p - 2} ε^{- (3 p - 2)} \int_{Σ_{t}} {| g_{_{_{f} r}} |}^{3 p - 2} d τ d γ . \end{matrix}

Due to the inequalities a. and b., from (28) we obtain

\begin{matrix} \frac{p_{_{1}}}{3 p - 2} [\int_{Ω} {| U (s, x) |}^{3 p - 2} d x + \int_{Γ} {| ζ (s, x) |}^{3 p - 2} d γ] \\ + λ p_{_{2}} \int_{Q_{s}} K (τ, x, U) \nabla U \cdot \nabla ({| U |}^{3 p - 4} U) d τ d x + (1 - λ) 3 (p - 1) p_{_{2}} \int_{Q_{s}} {| \nabla U |}^{2} {| U |}^{3 p - 4} d τ d x \\ + λ p_{_{r}} \int_{Q_{s}} {| U (τ, x) |}^{3 p} d τ d x \\ + p_{_{t}} \int_{Σ_{s}} {| ζ (τ, x) |}^{3 p - 2} d τ d γ + \int_{Σ_{s}} \nabla_{_{Γ}} ({| ζ |}^{3 p - 3}) \cdot \nabla_{_{Γ}} ζ d τ d γ \\ \leq \frac{p_{_{1}}}{3 p - 2} [\int_{Ω} | U_{0} {(x) |}^{3 p - 2} d x + \int_{Γ} {| ζ_{0} (x) |}^{3 p - 2} d γ] \\ + [λ p_{_{r}} + \frac{(3 p - 2) - 1}{3 p - 2} ε^{\frac{3 p - 2}{3 p - 3}}] \int_{Q_{s}} {| U (τ, x) |}^{3 p - 2} d τ d x \\ + \frac{(3 p - 2) - 1}{3 p - 2} ε^{\frac{3 p - 2}{3 p - 3}} \int_{Σ_{s}} {| U (τ, x) |}^{3 p - 2} d τ d x \\ + p_{_{s}} \frac{1}{3 p - 2} ε^{- (3 p - 2)} ∥ g_{_{d}} ∥_{L^{3 p - 2} (Q_{s})}^{3 p - 2} + \frac{1}{3 p - 2} ε^{- (3 p - 2)} {∥ g_{_{_{f} r}} ∥}_{L^{3 p - 2}}^{3 p - 2} (Σ_{s}) \end{matrix}

(29)

for all

s \in (0, T]

In particular, it follows that from (29) we obtain

\begin{matrix} \int_{Ω} {| U (s, x) |}^{3 p - 2} d x + \int_{Γ} {| ζ (s, x) |}^{3 p - 2} d γ \\ \leq C_{0} [∥ U_{0} {(x) ∥}_{L^{3 p - 2} (Ω)}^{3 p - 2} + ∥ ζ_{0} {(x) ∥}_{L^{3 p - 2} (Γ)}^{3 p - 2} + ∥ g_{_{d}} ∥_{L^{3 p - 2} (Q_{s})}^{3 p - 2} + {∥ g_{_{_{f} r}} ∥}_{L^{3 p - 2} (Σ_{s})}^{3 p - 2}] \\ + C_{0} \int_{0}^{t} [\int_{Ω} {| U (τ, x) |}^{3 p - 2} d τ d x + \int_{Γ} {| ζ (τ, x) |}^{3 p - 2} d γ] d τ \end{matrix}

(30)

where

C_{0} = C (| Ω |, | Γ |, p, p_{_{1}}, p_{_{2}}, p_{_{r}}, p_{_{t}}, p_{_{s}})

, in conjuction with (24)

_{2}

By Gronwall’s lemma and owing to

L^{3 p - 2} (Q) \subset L^{p} (Q)

, from (30) we obtain

\begin{matrix} {∥ U ∥}_{L^{p} (Q)}^{p} + {∥ ζ ∥}_{L^{p} (Σ)}^{p} \\ \leq C (T, C_{0}) [{∥ U ∥}_{L^{3 p - 2} (Q)}^{3 p - 2} + {∥ ζ ∥}_{L^{3 p - 2} (Σ)}^{3 p - 2}] \\ \leq C (T, C_{0}) [∥ U_{0} {(x) ∥}_{L^{3 p - 2} (Ω)}^{3 p - 2} + ∥ ζ_{0} {(x) ∥}_{L^{3 p - 2} (Γ)}^{3 p - 2} + ∥ g_{_{d}} ∥_{L^{3 p - 2} (Q)}^{3 p - 2} + {∥ g_{_{_{f} r}} ∥}_{L^{3 p - 2} (Σ)}^{3 p - 2}] . \end{matrix}

(31)

Having established an estimate for

{∥ U ∥}_{L^{3 p - 2} (Q)}^{3 p - 2} + {∥ ζ ∥}_{L^{3 p - 2} (Σ)}^{3 p - 2}

(see (31)), we now return to the relation in (29) to derive the following estimate:

\begin{matrix} λ p_{_{r}} {∥ | U |}^{3} ∥_{L^{p} (Q)}^{p} \\ \leq C (T, C_{0}) [∥ U_{0} {(x) ∥}_{L^{3 p - 2} (Ω)}^{3 p - 2} + ∥ ζ_{0} {(x) ∥}_{L^{3 p - 2} (Γ)}^{3 p - 2} + ∥ g_{_{d}} ∥_{L^{3 p - 2} (Q)}^{3 p - 2} + {∥ g_{_{_{f} r}} ∥}_{L^{3 p - 2} (Σ)}^{3 p - 2}], \end{matrix}

(32)

where the boundary condition in (24)

_{2}

is also used.

Applying Lemma 7.4 in Choban and Moroşanu [1] (p. 114) to the linear inhomogeneous problem (24) with

f_{3} = λ \{p_{_{r}} [U (t, x) - U^{3} (t, x)] + p_{_{s}} g_{_{d}} (t, x)\} \in L^{p} (Q)

and

g_{3} = λ g_{_{f r}} (t, x) \in L^{p} (Σ)

we obtain

\begin{matrix} {∥ U ∥}_{W_{p}^{1, 2} (Q)} + {∥ ζ ∥}_{W_{p}^{1, 2} (Σ)} \\ \leq C_{1} {∥ U_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Ω)} + ∥ ζ_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Γ)} + ∥ g_{_{d}} ∥_{L^{p} (Q)} + {∥ g_{_{_{f} r}} ∥}_{L^{p} (Σ)} \\ + λ p_{_{r}} [{∥ U ∥}_{L^{p} (Ω)} + {∥ | U |}^{3} ∥_{L^{p} (Ω)}]}, \end{matrix}

(33)

for a constant

C_{1} = C (n, C (T, C_{0})) > 0

Now using (31) and (32), (33) then becomes

\begin{matrix} {∥ U ∥}_{W_{p}^{1, 2} (Q)} + {∥ ζ ∥}_{W_{p}^{1, 2} (Σ)} \\ \leq C_{1} {1 + ∥ U_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Ω)} + ∥ ζ_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Γ)} + ∥ U_{0} ∥_{L^{3 p - 2} (Ω)}^{\frac{3 p - 2}{p}} + {∥ ζ_{0} ∥}_{L^{3 p - 2} (Γ)}^{\frac{3 p - 2}{p}} \\ + ∥ g_{_{d}} ∥_{L^{3 p - 2} (Q)}^{\frac{3 p - 2}{p}} + ∥ g_{_{_{f} r}} ∥_{L^{3 p - 2} (Σ)}^{\frac{3 p - 2}{p}} + ∥ g_{_{d}} ∥_{L^{p} (Q)} + {∥ g_{_{_{f} r}} ∥}_{L^{p} (Σ)}}, \end{matrix}

(34)

The inclusions in (20) guarantee that

{∥ U ∥}_{L^{p} (Q)} + {∥ ζ ∥}_{L^{p} (Σ)} \leq C ({∥ U ∥}_{W_{p}^{1, 2} (Q)} + {∥ ζ ∥}_{W_{p}^{1, 2} (Σ)})

where, thanks to (34), we may conclude that a constant

R > 0

exists such that the property in (23) is true.

Denoting

B_{R}^{H} : = \{(U, ζ) \in {B : ∥ (U, ζ) ∥}_{B} < R\},

relation (23) implies that

(U, ζ, λ) \neq (U, ζ) \forall (U, ζ) \in \partial B_{R}^{H}, \forall λ \in [0, 1],

provided that

R > 0

is sufficiently large. Furthermore, following the same ideas in [1,3,4,5,6,7,16,20], we can conclude that problem (8) has the solution

(U, ζ) \in W_{p}^{1, 2} (Q) \times W_{p}^{1, 2} (Σ)

Making use of the embedded

L^{3 p - 2} (Q) \subset L^{p} (Q)

and the estimate (34), it follows that (11) and this completes the proof of the first part in Theorem 1.

Proof of Theorem 1 Continued

In this subsection we demonstrate the second part of Theorem 1 which entails checking (14) and thus the uniqueness of the solution to (1) (or (3)). We consider (

U^{1}, ζ^{1}

) and (

U^{2}, ζ^{2}

) as in the statement of Theorem 1. From the first part we know that

U^{1}, U^{2} \in W_{p}^{1, 2} (Q)

and

ζ^{1}, ζ^{2} \in W_{p}^{1, 2} (Σ)

. Therefore,

U = U^{1} - U^{2} \in W_{p}^{1, 2} (Q)

and

Z = ζ^{1} - ζ^{2} \in W_{p}^{1, 2} (Σ)

Following [1,2,3,5,6,7,16,20], the increments of

a_{i j}

and A (see (4)) can be written in the following form

a_{i j} (s, x, U^{1}, U_{x}^{1}) - a_{i j} (s, x, U^{2}, U_{x}^{2}) = \int_{0}^{1} \frac{d}{d λ} a_{i, j} (s, x, U^{λ}, U_{x}^{λ}) d λ,

A (s, x, U^{1}, U_{x}^{1}) - A (s, x, U^{2}, U_{x}^{2}) = \int_{0}^{1} \frac{d}{d λ} A (s, x, U^{λ}, U_{x}^{λ}) d λ

and so

\begin{matrix} a_{i j} (s, x, U^{1}, U_{x}^{1}) U_{_{x_{i} x_{j}}}^{1} - a_{i j} (s, x, U^{2}, U_{x}^{2}) U_{_{x_{i} x_{j}}}^{2} \\ = a_{i j} (s, x, U^{1}, U_{x}^{1}) U_{_{x_{i} x_{j}}} + \{U_{_{x_{i} x_{j}}}^{2} \int_{0}^{1} \frac{\partial}{\partial U_{x_{j}}^{λ}} a_{i, j} (s, x, U^{λ}, U_{x}^{λ}) d λ\} U_{x_{i}}, \end{matrix}

(35)

\begin{matrix} A (s, x, U^{1}, U_{x}^{1}) - A (s, x, U^{2}, U_{x}^{2}) = \{\int_{0}^{1} \frac{\partial}{\partial U_{x_{j}}^{λ}} A (s, x, U^{λ}, U_{x}^{λ}) d λ\} U_{x_{i}}, \end{matrix}

(36)

where

a_{i, j} (s, x, U_{x}^{λ}, U_{x}^{λ}) = \frac{\partial}{\partial U_{x_{j}}^{λ}} [K (s, x, U^{λ}) U_{x_{i}}^{λ}]

A (s, x, U^{λ}, U_{x}^{λ}) = a_{i} (s, x, U^{λ}, U_{x}^{λ})

a_{i} (s, x, U^{λ}, U_{x}^{λ}) = \frac{\partial}{\partial x_{i}} [K (s, x, U^{λ}) U_{x_{i}}^{λ}]

U^{λ} (s, x) = λ U^{1} (s, x) + (1 - λ) U^{2} (s, x)

and

U_{x}^{λ} (s, x) = λ U_{x}^{1} (s, x) + (1 - λ) U_{x}^{2} (s, x)

Subtracting (3) for

U^{2} (s, x)

from (3) for

U^{1} (s, x)

and using (35) and (36), we obtain the following linear parabolic problem with inhomogeneous dynamic boundary conditions, i.e.,

\{\begin{matrix} p_{_{1}} \frac{\partial}{\partial t} U - {\hat{a}}_{i j} (s, x) Δ U = - {\hat{a}}_{i} (s, x) \nabla U - p_{_{2}} U + p_{_{s}} (g_{_{d}}^{1} - g_{_{d}}^{2}) & in Q \\ U (s, x) = Z (s, x) & on Σ \\ U (0, x) = (U_{0}^{1} - U_{0}^{2}) (x) & in Ω \\ p_{_{1}} \frac{\partial}{\partial n} U + p_{_{2}} \frac{\partial}{\partial t} Z - Δ_{Γ} Z + p_{_{t}} Z = g_{_{_{f} r}}^{1} - g_{_{_{f} r}}^{2} & on Σ \\ Z (0, x) = (ζ_{0}^{1} - ζ_{0}^{2}) (x) & on Γ, \end{matrix}

(37)

where

{\hat{a}}_{i j} (s, x) = a_{i j} (s, x, U^{1}, U_{x}^{1}),

{\hat{a}}_{i} (s, x) = - U_{_{x_{i} x_{j}}}^{2} \int_{0}^{1} \frac{\partial}{\partial U_{x_{j}}^{λ}} a_{i, j} (s, x, U^{λ}, U_{x}^{λ}) d λ

+ \int_{0}^{1} \frac{\partial}{\partial U_{x_{j}}^{λ}} \frac{\partial}{\partial x_{i}} [K (s, x, U^{λ}) U_{x_{i}}^{λ}] d λ .

Next, following the work of A. Miranville and C. Moroşanu [3], we easily deduce the validity of the estimate in (14); thus, the uniqueness of the solution to (1) or (3) is true. □

Corollary 1.

Corresponding to

U_{0}^{1} = U_{0}^{2}

and

ζ_{0}^{1} = ζ_{0}^{2}

, the problem (1) possesses a unique classical solution.

4. Approximating Scheme—Convergence and Error Estimate

Here we use the fractional steps method in order to approximate the unique solution to problem (8) with inhomogeneous dynamic boundary conditions (see Corollary 1). Precisely,

\forall ε > 0

, let

M_{ε} = [\frac{T}{ε}]

and

Q_{i}^{ε} = [i ε, (i + 1) ε] \times Ω, Σ_{i}^{ε} = [i ε, (i + 1) ε] \times \partial Ω i = 0, 1, \dots, M_{ε} - 1,

with

Q_{M_{ε} - 1}^{ε} = [(M_{ε} - 1) ε, T] \times Ω

Σ_{M_{ε} - 1}^{ε} = [(M_{ε} - 1) ε, T] \times \partial Ω

. Correspondingly, we link the following numerical scheme with problem (8)

\{\begin{matrix} p_{_{1}} \frac{\partial}{\partial t} U^{ε} - p_{_{2}} div (K (t, x, U^{ε}) \nabla U^{ε}) = p_{_{r}} U^{ε} + p_{_{s}} g_{_{d}} (t, x) & in Q_{i}^{ε} \\ p_{_{2}} \frac{\partial}{\partial n} U^{ε} + p_{_{1}} \frac{\partial}{\partial t} ζ^{ε} - Δ_{_{Γ}} ζ^{ε} + p_{_{t}} ζ^{ε} = g_{_{f r}} (t, x) & on Σ_{i}^{ε} \\ U^{ε} (i ε, x) = z (ε, U_{-}^{ε} (i ε, x)) & on Ω \\ ζ^{ε} (i ε, x) = U^{ε} (i ε, x) & on \partial Ω, \end{matrix}

(38)

with

z (ε, U_{-}^{ε} (i ε, x))

being the solution of Cauchy problem:

\{\begin{matrix} z^{'} (s) + p_{_{r}} z^{3} (s) = 0 & s \in [0, ε] \\ z (0) = U_{-}^{ε} (i ε, x) & on Ω \\ U_{-}^{ε} (0, x) = U_{0} (x) & on Ω \\ U_{-}^{ε} (0, x) = ζ_{0} (x) & on \partial Ω, \end{matrix}

(39)

where

U_{-}^{ε}

stands for the left-hand limit of

U^{ε}

For a detailed discussion regarding the importance of the above numerical scheme we direct the reader to the works [5,9,11,12,13,14,17,18,19,22,23].

The main question of this work concerns the convergence as

ε \to 0

of the sequence

(U^{ε}, ζ^{ε})

of the solutions to problems (38) and (39), and to the solution

(U, ζ)

of problem (8) (see [11] for more details).

For simplicity, we note:

W_{_{Q}} = L^{2} ([0, T]; H^{1} (Ω)) \cap L^{\infty} (Q)

and

W_{_{Σ}} = L^{2} ([0, T]; H^{1} (\partial Ω)) \cap L^{\infty} (Σ)

Definition 2.

By a weak solution to problem (8) we refer to a pair of functions

(U, ζ) \in W_{_{Q}} \times W_{_{Σ}}

and

U = ζ

on Σ, which satisfy (8) in the following sense:

\begin{matrix} p_{_{1}} \int_{Q} (\frac{\partial}{\partial t} U, ϕ_{1}) d t d x + p_{_{2}} \int_{Q} K (t, x, U) \nabla U \cdot \nabla ϕ_{1} d t d x \\ + p_{_{2}} \int_{Σ} (\frac{\partial}{\partial t} ζ, ϕ_{2}) d t d γ + \int_{Σ} \nabla ζ \nabla ϕ_{2} d t d γ + p_{_{t}} \int_{Σ} ζ ϕ_{2} d t d γ \\ = p_{_{r}} \int_{Q} (U - U^{3}) ϕ_{1} d t d x + p_{_{s}} \int_{Q} g_{_{d}} ϕ_{1} d t d x + \int_{Σ} g_{_{f r}} ϕ_{2} d t d γ \\ \forall (ϕ_{1}, ϕ_{2}) \in L^{2} ([0, T]; H^{1} (Ω)) \times L^{2} ([0, T]; H^{1} (Γ)), \end{matrix}

(40)

where

ϕ_{1} = ϕ_{2}

on Σ, and

U (0, x) = U_{0} (x)

on Ω.

Definition 3.

By a weak solution to problems (38) and (39) we refer to a pair of functions

(U^{ε}, ζ^{ε}) \in W_{Q_{i}^{ε}} \times W_{Σ_{i}^{ε}}

, and

U_{i}^{ε} = ζ_{i}^{ε}

Σ_{i}^{ε}

i \in {0, 1, \dots, M_{ε} - 1}

, which satisfy (38) and (39) in the following sense:

\begin{matrix} p_{_{1}} \int_{Q} (\frac{\partial}{\partial t} U^{ε}, ξ_{1}) d t d x + p_{_{2}} \int_{Q} K (t, x, U^{ε}) \nabla U^{ε} \cdot \nabla ξ_{1} d t d x \\ + p_{_{2}} \int_{Σ} (\frac{\partial}{\partial t} ζ^{ε}, ξ_{2}) d t d γ + \int_{Σ} \nabla ζ^{ε} \nabla ξ_{2} d t d γ + p_{_{t}} \int_{Σ} ζ^{ε} ξ_{2} d t d γ \\ = p_{_{r}} \int_{Q} U^{ε} ξ_{1} d t d x + p_{_{s}} \int_{Q} g_{_{d}} ξ_{1} d t d x + \int_{Σ} g_{_{f r}} ξ_{2} d t d γ \\ \forall (ξ_{1}, ξ_{2}) \in L^{2} ([0, T]; H^{1} (Ω)) \times L^{2} ([0, T]; H^{1} (\partial Ω)), \end{matrix}

(41)

where

U_{-}^{ε} (0, x) = U_{0} (x)

on Ω, and

U_{-}^{ε} (0, x) = ζ_{0} (x)

\partial Ω

In (40) and (41) the symbols

\int_{Q}

and

\int_{Σ}

denote the duality between

L^{2} ([0, T]; H^{1} (Ω))

and

L^{2} ([0, T]; H^{1} {(Ω)}^{'})

as well as

L^{2} ([0, T]; H^{1} (\partial Ω))

and

L^{2} ([0, T]; H^{1} {(\partial Ω)}^{'})

, respectively.

Convergence of the Numerical Schemes (38) and (39)

The purpose of this subsection is to prove the convergence of the solution to the numerical scheme associated with the non-linear problem (8). Therefore,

Theorem 2.

Assume that

U_{0} (x) \in W_{\infty}^{2 - \frac{2}{2}} (Ω)

, satisfying

p_{_{2}} \frac{\partial}{\partial ν} U_{0} - Δ_{_{Γ}} U_{0} + p_{_{t}} U_{0} = g_{_{f r}} (0, x)

\partial Ω

and

g_{_{f r}} (s, x) \in W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)

. Let

(U^{ε}, ζ^{ε})

be the solution to the numerical schemes (38) and (39). As

ε \to 0

, one has

(U^{ε}, ζ^{ε}) \to (U^{★}, ζ^{★}) strongly in L^{2} (Ω) \times L^{2} (\partial Ω) for any s \in (0, T],

(42)

where

(U^{★}, ζ^{★}) \in L^{2} ([0, T]; H^{1} (Ω)) \times L^{2} ([0, T]; H^{1} (\partial Ω))

is a weak solution to problem(8).

The following lemmas, which involve the Cauchy problem (39), are very useful in the proof of Theorem 2. These were proven for the first time in [11]. Here, we reproduce them as well as sketch out the proof when pertinent.

Lemma 1.

Assume

U_{-}^{ε} (i ε, x) \in L^{\infty} (Ω)

i = 0, 1, \dots, M_{ε} - 1

. Then,

U^{ε} (i ε, x) \in L^{\infty} (Ω)

and

∥ U^{ε} {(i ε, x) ∥}_{L^{2} (Ω)}^{2} \leq {∥ U_{-}^{ε} (i ε, x) ∥}_{L^{2} (Ω)}^{2} .

(43)

Proof.

We write (39)

_{1}

in the form

{(\frac{1}{z^{2}})}^{'} = p_{_{r}}

, and following the same reasoning as in [11]

we obtain

z^{2} (ε, U_{-}^{ε} (i ε, x)) \leq U_{-}^{ε} {(i ε, x)}^{2}, a . e x \in Ω .

(44)

Owing to (38)

_{3}

and (44), we can easily conclude the inequality complete in (43). □

Lemma 2.

For

i = 0, 1, \dots, M_{ε} - 1

, the estimate below holds

∥ \nabla U^{ε} {(i ε, x) ∥}_{L^{2} (Ω)} \leq {∥ \nabla U_{-}^{ε} (i ε, x) ∥}_{L^{2} (Ω)} .

(45)

Lemma 3.

The following estimate holds

∥ z (ε, x) - U_{-}^{ε} {(i ε, x) ∥}_{L^{2} (Ω)} \leq ε L

(46)

where

L > 0

depends on

| Ω |

∥ U_{-}^{ε} ∥_{L^{\infty} (Ω)}

and

p_{_{2}}

Now, we are in a position to give the proof of Theorem 2. Following the same steps as in [11], we obtain the solution to problem (38) as

(U^{ε}, ζ^{ε}) \in W_{p}^{1, 2} (Q_{i}^{ε}) \cap L^{\infty} (Q_{i}^{ε}) \times W_{p}^{1, 2} (Σ_{i}^{ε}) \cap L^{\infty} (Σ_{i}^{ε})

\forall i \in {0, 1, \dots, M_{ε} - 1}

Next, we give a priori estimates to

Q_{i}^{ε}, \forall i \in {0, 1, \dots, M_{ε} - 1}

. Firstly, we multiply (38)

_{1}

U_{t}^{ε}

and obtain

\begin{matrix} p_{_{1}} \int_{Ω} | U_{t}^{ε} |^{2} d x + p_{_{1}} \int_{Γ} {| ζ_{t}^{ε} |}^{2} d γ \\ + \frac{p_{_{2}}}{2} \int_{Ω} K (t, x, U^{ε}) \frac{d}{d t} | \nabla U^{ε} |^{2} d x + \frac{1}{2} \frac{d}{d t} \int_{Γ} | \nabla_{Γ} ζ^{ε} |^{2} d γ + \frac{p_{_{t}}}{2} \frac{d}{d t} \int_{Γ} {| ζ^{ε} |}^{2} d γ \\ = \frac{p_{_{2}}}{2} \frac{d}{d t} \int_{Ω} {| U^{ε} |}^{2} d x + \int_{Γ} g_{_{f r}} ζ_{t}^{ε} d γ + p_{_{s}} \int_{Ω} g_{_{d}} U_{t}^{ε} d x . \end{matrix}

(47)

Using Hölder’s inequality for the right-hand terms

\int_{Γ} g_{_{f r}} ζ_{t}^{ε} d γ

and

\int_{Ω} g_{_{d}} U_{t}^{ε} d x

, we have

\int_{Γ} g_{_{f r}} ζ_{t}^{ε} d γ \leq \frac{p_{_{1}}}{2} \int_{Γ} | ζ_{t}^{ε} |^{2} d γ + \frac{1}{2 p_{_{1}}} \int_{Γ} {| g_{_{f r}} |}^{2} d γ,

p_{_{s}} \int_{Ω} g_{_{d}} U_{t}^{ε} d x \leq \frac{p_{_{1}}}{2} \int_{Ω} | U_{t}^{ε} |^{2} d x + \frac{p_{_{s}}}{2 p_{_{1}}} \int_{Ω} {| g_{_{d}} |}^{2} d x,

and substituting them in (47), we derive

\begin{matrix} \frac{p_{_{1}}}{2} \int_{Ω} | U_{t}^{ε} |^{2} d x + \frac{p_{_{1}}}{2} \int_{Γ} {| ζ_{t}^{ε} |}^{2} d γ \\ + \frac{p_{_{2}}}{2} K_{m i n} \frac{d}{d t} \int_{Ω} | \nabla U^{ε} |^{2} d x + \frac{1}{2} \frac{d}{d t} \int_{Γ} | \nabla_{Γ} ζ^{ε} |^{2} d γ + \frac{p_{_{t}}}{2} \frac{d}{d t} \int_{Γ} {| ζ^{ε} |}^{2} d γ \\ \leq \frac{p_{_{2}}}{2} \frac{d}{d t} \int_{Ω} | U^{ε} |^{2} d x + \frac{1}{2 p_{_{1}}} \int_{Γ} | g_{_{f r}} |^{2} d γ + \frac{p_{_{s}}}{2 p_{_{1}}} \int_{Ω} {| g_{_{d}} |}^{2} d x, \end{matrix}

(48)

where the inequality (9) is also used.

Multiplying (38)

_{1}

\frac{1}{p_{_{1}} p_{_{2}}} U^{ε}

as shown above, we obtain

\begin{matrix} \frac{1}{2 p_{_{2}}} \frac{d}{d t} \int_{Ω} | U^{ε} |^{2} d x + \frac{1}{2 p_{_{2}}} \frac{d}{d t} \int_{Γ} {| ζ^{ε} |}^{2} d γ \\ + \frac{1}{p_{_{1}}} \int_{Ω} K (t, x, U^{ε}) | \nabla U^{ε} |^{2} d x + \frac{1}{p_{_{1}}} \int_{Γ} | \nabla_{Γ} ζ^{ε} |^{2} d γ + \frac{p_{_{t}}}{p_{_{1}} p_{_{2}}} \int_{Γ} {| ζ^{ε} |}^{2} d γ \\ = \frac{1}{p_{_{1}} p_{_{2}} p_{_{r}}} \int_{Ω} {| U^{ε} |}^{2} d x + \frac{1}{p_{_{1}} p_{_{2}}} \int_{Γ} g_{_{f r}} ζ^{ε} d γ + \frac{p_{_{s}}}{p_{_{1}} p_{_{2}}} \int_{Ω} g_{_{d}} U^{ε} d x . \end{matrix}

(49)

In addition, using Hölder’s inequality for the right-hand terms

\int_{Γ} g_{_{f r}} ζ^{ε} d γ

and

\int_{Ω} g_{_{d}} U^{ε} d x

, we have

\frac{1}{p_{_{1}} p_{_{2}}} \int_{Γ} g_{_{f r}} ζ^{ε} d γ \leq \frac{2 p_{_{t}}}{p_{_{1}} p_{_{2}}} \int_{Γ} | ζ^{ε} |^{2} d γ + \frac{1}{2 p_{_{t}} p_{_{1}} p_{_{2}}} \int_{Γ} {| g_{_{f r}} |}^{2} d γ,

\frac{p_{_{s}}}{p_{_{1}} p_{_{2}}} \int_{Ω} g_{_{d}} U^{ε} d x \leq \frac{1}{p_{_{1}} p_{_{2}}} \int_{Ω} | U^{ε} |^{2} d x + \frac{p_{_{s}}}{p_{_{1}} p_{_{2}}} \int_{Ω} {| g_{_{d}} |}^{2} d x,

and then from (49) we obtain

\begin{matrix} \frac{1}{2 p_{_{2}}} \frac{d}{d t} \int_{Ω} | U^{ε} |^{2} d x + \frac{1}{2 p_{_{2}}} \frac{d}{d t} \int_{Γ} {| ζ^{ε} |}^{2} d γ \\ + \frac{1}{p_{_{1}}} K_{m i n} \int_{Ω} | \nabla U^{ε} |^{2} d x + \frac{1}{p_{_{1}}} \int_{Γ} {| \nabla_{Γ} ζ^{ε} |}^{2} d γ \\ \leq C (p_{_{s}}, p_{_{t}}, p_{_{1}}, p_{_{2}}) [\int_{Ω} | U^{ε} |^{2} d x + \int_{Γ} | ζ^{ε} |^{2} d γ + \int_{Γ} | g_{_{f r}} |^{2} d γ + \int_{Ω} {| g_{_{d}} |}^{2} d x], \end{matrix}

(50)

where the inequality (9) is also used.

Adding (48) and (50), we obtain

\begin{matrix} \frac{\partial}{\partial t} [\frac{1}{2 p_{_{2}}} \int_{Ω} | U^{ε} |^{2} d x + (\frac{p_{_{t}}}{2} + \frac{1}{2 p_{_{2}}}) \int_{Γ} | ζ^{ε} |^{2} d γ + \frac{p_{_{2}}}{2} K_{m i n} \int_{Ω} | \nabla U^{ε} |^{2} d x + \frac{1}{2} \int_{Γ} {| \nabla_{Γ} ζ^{ε} |}^{2} d x] \\ + \frac{p_{_{1}}}{2} \int_{Ω} | U_{t}^{ε} |^{2} d x + \frac{p_{_{1}}}{2} \int_{Γ} | ζ_{t}^{ε} |^{2} d γ + \frac{K_{m i n}}{p_{_{1}}} \int_{Ω} | \nabla U^{ε} |^{2} d x + \frac{1}{p_{_{1}}} \int_{Γ} {| \nabla_{Γ} ζ^{ε} |}^{2} d γ \\ \leq C (p_{_{s}}, p_{_{t}}, p_{_{1}}, p_{_{2}}) [\int_{Ω} | U^{ε} |^{2} d x + \int_{Γ} | ζ^{ε} |^{2} d γ + \int_{Γ} | g_{_{f r}} |^{2} d γ + \int_{Ω} {| g_{_{d}} |}^{2} d x] . \end{matrix}

Integrating the preceding on

Q_{0}^{ε},

we derive

\begin{matrix} \frac{1}{2 p_{_{2}}} ∥ U_{-}^{ε} {(ε, x) ∥}_{L^{2} (Ω)}^{2} + (\frac{p_{_{t}}}{2} + \frac{1}{2 p_{_{2}}}) {∥ ζ_{-}^{ε} (ε, x) ∥}_{L^{2} (Γ)}^{2} \\ + \frac{p_{_{2}}}{2} K_{m i n} ∥ \nabla U_{-}^{ε} {(ε, x) ∥}_{L^{2} (Ω)}^{2} + \frac{1}{2} {∥ \nabla_{Γ} ζ_{-}^{ε} (ε, x) ∥}_{L^{2} (Γ)}^{2} \\ + \int_{0}^{ε} [\frac{p_{_{1}}}{2} \int_{Ω} | U_{t}^{ε} |^{2} d x + \frac{p_{_{1}}}{2} \int_{Γ} | ζ_{t}^{ε} |^{2} d γ + \frac{K_{m i n}}{p_{_{1}}} \int_{Ω} | \nabla U^{ε} |^{2} d x + \frac{1}{p_{_{1}}} \int_{Γ} {| \nabla_{Γ} ζ^{ε} |}^{2} d γ] d s \\ \leq \frac{1}{2 p_{_{2}}} ∥ U_{0} ∥_{L^{2} (Ω)}^{2} + (\frac{p_{_{t}}}{2} + \frac{1}{2 p_{_{2}}}) ∥ ζ_{0} ∥_{L^{2} (Γ)}^{2} + \frac{p_{_{2}}}{2} K_{m i n} ∥ \nabla U_{0} ∥_{L^{2} (Ω)}^{2} + \frac{1}{2} {∥ \nabla_{Γ} ζ_{0} ∥}_{L^{2} (Γ)}^{2} \\ + C (p_{_{s}}, p_{_{t}}, p_{_{1}}, p_{_{2}}) \{\int_{0}^{ε} [∥ U^{ε} ∥_{L^{2} (Ω)}^{2} + {∥ ζ^{ε} ∥}_{L^{2} (Γ)}^{2}] d s + ∥ g_{_{f r}} ∥_{L^{2} (Σ_{0}^{ε})}^{2} + {∥ g_{_{d}} ∥}_{L^{2} (Q_{0}^{ε})}^{2}\} . \end{matrix}

(51)

It is relatively easy to observe that the estimate above refers to

Q_{0}^{ε}

and

Σ_{0}^{ε}

(

i = 0

). Proceeding in a similar way for

i = 1, 2, \dots, M_{ε} - 2

, we obtain

\begin{matrix} \frac{1}{2 p_{_{2}}} ∥ U_{-}^{ε} {((i + 1) ε, x) ∥}_{L^{2} (Ω)}^{2} + (\frac{p_{_{t}}}{2} + \frac{1}{2 p_{_{2}}}) {∥ ζ_{-}^{ε} ((i + 1) ε, x) ∥}_{L^{2} (Γ)}^{2} \\ + \frac{p_{_{2}}}{2} K_{m i n} ∥ \nabla U_{-}^{ε} {((i + 1) ε, x) ∥}_{L^{2} (Ω)}^{2} + \frac{1}{2} {∥ \nabla_{Γ} ζ_{-}^{ε} ((i + 1) ε, x) ∥}_{L^{2} (Γ)}^{2} \\ + \int_{i ε}^{(i + 1) ε} [\frac{p_{_{1}}}{2} ∥ U_{t}^{ε} ∥_{L^{2} (Ω)}^{2} + \frac{p_{_{1}}}{2} ∥ ζ_{t}^{ε} ∥_{L^{2} (Γ)}^{2} + \frac{K_{m i n}}{p_{_{1}}} ∥ \nabla U^{ε} ∥_{L^{2} (Ω)}^{2} + \frac{1}{p_{_{1}}} {∥ \nabla_{Γ} ζ^{ε} ∥}_{L^{2} (Γ)}^{2}] d s \\ \leq \frac{1}{2 p_{_{2}}} ∥ U^{ε} {(i ε, x) ∥}_{L^{2} (Ω)}^{2} + (\frac{p_{_{t}}}{2} + \frac{1}{2 p_{_{2}}}) {∥ ζ^{ε} (i ε, x) ∥}_{L^{2} (Γ)}^{2} \\ + \frac{p_{_{2}}}{2} ∥ \nabla U^{ε} {(i ε, x) ∥}_{L^{2} (Ω)}^{2} + \frac{1}{2} {∥ \nabla_{Γ} ζ^{ε} (i ε, x) ∥}_{L^{2} (Γ)}^{2} \\ + C (p_{_{s}}, p_{_{t}}, p_{_{1}}, p_{_{2}}) \{\int_{i ε}^{(i + 1) ε} [∥ U^{ε} ∥_{L^{2} (Ω)}^{2} + {∥ ζ^{ε} ∥}_{L^{2} (Γ)}^{2}] d s + ∥ g_{_{f r}} ∥_{L^{2} (Σ_{i}^{ε})}^{2} + {∥ g_{_{d}} ∥}_{L^{2} (Q_{i}^{ε})}^{2}\}, \end{matrix}

(52)

while for

i = M_{ε} - 1

we have

\begin{matrix} \frac{1}{2 p_{_{2}}} ∥ U_{-}^{ε} {(T, x) ∥}_{L^{2} (Ω)}^{2} + (\frac{p_{_{t}}}{2} + \frac{1}{2 p_{_{2}}}) {∥ ζ_{-}^{ε} (T, x) ∥}_{L^{2} (Γ)}^{2} \\ + \frac{p_{_{2}}}{2} K_{m i n} ∥ \nabla U_{-}^{ε} {(T, x) ∥}_{L^{2} (Ω)}^{2} + \frac{1}{2} {∥ \nabla_{Γ} ζ_{-}^{ε} (T, x) ∥}_{L^{2} (Γ)}^{2} \\ + \int_{M_{ε} - 1}^{T} [\frac{p_{_{1}}}{2} ∥ U_{t}^{ε} ∥_{L^{2} (Ω)}^{2} + \frac{p_{_{1}}}{2} ∥ ζ_{t}^{ε} ∥_{L^{2} (Γ)}^{2} + \frac{1}{p_{_{1}}} ∥ \nabla U^{ε} ∥_{L^{2} (Ω)}^{2} + \frac{1}{p_{_{1}}} {∥ \nabla_{Γ} ζ^{ε} ∥}_{L^{2} (Γ)}^{2}] d s \\ \leq \frac{1}{2 p_{_{2}}} ∥ U^{ε} {(T, x) ∥}_{L^{2} (Ω)}^{2} + (\frac{p_{_{t}}}{2} + \frac{1}{2 p_{_{2}}}) {∥ ζ^{ε} (T, x) ∥}_{L^{2} (Γ)}^{2} \\ + \frac{p_{_{2}}}{2} ∥ \nabla U^{ε} {(T, x) ∥}_{L^{2} (Ω)}^{2} + \frac{1}{2} {∥ \nabla_{Γ} ζ^{ε} (T, x) ∥}_{L^{2} (Γ)}^{2} \\ + C (p_{_{s}}, p_{_{t}}, p_{_{1}}, p_{_{2}}) \{\int_{M_{ε} - 1}^{T} [∥ U^{ε} ∥_{L^{2} (Ω)}^{2} + {∥ ζ^{ε} ∥}_{L^{2} (Γ)}^{2}] d s + ∥ g_{_{f r}} ∥_{L^{2} (Σ_{M_{ε} - 1}^{ε})}^{2} + {∥ g_{_{d}} ∥}_{L^{2} (Q_{M_{ε} - 1}^{ε})}^{2}\} . \end{matrix}

(53)

Adding (51)–(53) and owing to the inequalities (43) and (45), we obtain

\begin{matrix} \frac{1}{2 p_{_{2}}} ∥ U_{-}^{ε} {(T, x) ∥}_{L^{2} (Ω)}^{2} + (\frac{p_{_{t}}}{2} + \frac{1}{2 p_{_{2}}}) {∥ ζ_{-}^{ε} (T, x) ∥}_{L^{2} (Γ)}^{2} \\ + \frac{p_{_{2}}}{2} ∥ \nabla U_{-}^{ε} {(T, x) ∥}_{L^{2} (Ω)}^{2} + \frac{1}{2} {∥ \nabla_{Γ} ζ_{-}^{ε} (T, x) ∥}_{L^{2} (Γ)}^{2} \\ + \int_{0}^{T} [\frac{p_{_{1}}}{2} ∥ U_{t}^{ε} ∥_{L^{2} (Ω)}^{2} + \frac{p_{_{1}}}{2} ∥ ζ_{t}^{ε} ∥_{L^{2} (Γ)}^{2} + \frac{1}{p_{_{1}}} ∥ \nabla U^{ε} ∥_{L^{2} (Ω)}^{2} + \frac{1}{p_{_{1}}} {∥ \nabla_{Γ} ζ^{ε} ∥}_{L^{2} (Γ)}^{2}] d t \\ \leq \frac{1}{2 p_{_{2}}} ∥ U_{0} ∥_{L^{2} (Ω)}^{2} + (\frac{p_{_{t}}}{2} + \frac{1}{2 p_{_{2}}}) ∥ ψ_{0} ∥_{L^{2} (Γ)}^{2} + \frac{p_{_{2}}}{2} ∥ \nabla U_{0} ∥_{L^{2} (Ω)}^{2} + \frac{1}{2} {∥ \nabla_{Γ} ζ_{0} ∥}_{L^{2} (Γ)}^{2} \\ + C (p_{_{s}}, p_{_{t}}, p_{_{1}}, p_{_{2}}) \{\int_{0}^{T} [∥ U^{ε} ∥_{L^{2} (Ω)}^{2} + {∥ ζ^{ε} ∥}_{L^{2} (Γ)}^{2}] d t + ∥ g_{_{f r}} ∥_{L^{2} (Σ)}^{2} + {∥ g_{_{d}} ∥}_{L^{2} (Q)}^{2}\} . \end{matrix}

Applying the Gronwall inequality to the above inequalities, we finally deduce

\int_{0}^{T} \{∥ U_{t}^{ε} ∥_{L^{2} (Ω)}^{2} + ∥ ζ_{t}^{ε} ∥_{L^{2} (Γ)}^{2} + ∥ \nabla U^{ε} ∥_{L^{2} (Ω)}^{2} + {∥ \nabla_{Γ} ζ^{ε} ∥}_{L^{2} (Γ)}^{2}\} d t \leq C,

(54)

where

C > 0

is independent of

ε

and

M_{ε}

Owing to (38)

_{3}

, (38)

_{4}

and (46), we obtain

\sum_{i = 0}^{M_{ε} - 1} {∥ U^{ε} (i ε, x) - U_{-}^{ε} (i ε, x) ∥}_{L^{2} (Ω)} \leq T L = C_{1},

(55)

\sum_{i = 0}^{M_{ε} - 1} {∥ ζ^{ε} (i ε, x) - ζ_{-}^{ε} (i ε, x) ∥}_{L^{2} (Γ)} \leq C_{2},

(56)

where

C_{1} > 0

and

C_{2} > 0

are independent of

M_{ε}

and

ε

. Summing (54)–(56), we derive

{V 1}_{0}^{T} U^{ε} + {V 2}_{0}^{T} ζ^{ε} + \int_{0}^{T} [∥ U_{t}^{ε} ∥_{L^{2} (Ω)}^{2} + ∥ ζ_{t}^{ε} ∥_{L^{2} (Γ)}^{2} + ∥ \nabla U^{ε} ∥_{L^{2} (Ω)}^{2} + {∥ \nabla_{_{Γ}} ζ^{ε} ∥}_{L^{2} (Γ)}^{2}] d s \leq C,

(57)

where the positive constant C is independent of

M_{ε}

and

ε

, while

{V 1}_{0}^{T} U^{ε}

and

{V 2}_{0}^{T} ζ^{ε}

stand for the variation of

U^{ε} : [0, T] \to L^{2} (Ω)

and

ζ^{ε} : [0, T] \to L^{2} (Γ)

, respectively.

Since the introduction of

L^{2} (Ω)

into

H^{- 1} (Ω)

is compact and

{U_{s}^{ε} (s)}

is bounded in

L^{2} (Ω)

\forall s \in [0, T]

, we conclude that there exists a bounded variation function

U^{*} (s) \in B V ([0, T]; H^{- 1} (Ω))

and subsequent

U^{ε} (s)

(see [11]), such that

U^{ε} (s) \to U^{*} (s) strongly in H^{- 1} (Ω) \forall s \in [0, T],

(58)

ζ^{ε} (s) \to ζ^{*} (s) strongly in H^{- 1} (Γ) \forall s \in [0, T] .

(59)

Further, from (57) we deduce that

\{\begin{matrix} U^{ε} \to U^{*} & w e a k l y i n L^{2} (0, T; H^{1} (Ω)) \\ ζ^{ε} \to ζ^{*} & w e a k l y i n L^{2} (0, T; H^{1} (Γ)) . \end{matrix}

(60)

By the well-known embeddings

H^{1} (Ω) \subset L^{2} (Ω) \subset H^{- 1} (Ω)

, and

H^{1} (\partial Ω) \subset L^{2} (\partial Ω) \subset H^{- 1} (\partial Ω)

, standard interpolation inequalities (see [11] p. 17) yield that

\forall ℓ > 0

\exists C (ℓ) > 0

such that

\{\begin{matrix} ∥ U^{ε} (s) - U^{*} {(s) ∥}_{L^{2} (Ω)} \leq ℓ ∥ U^{ε} (s) - U^{*} {(s) ∥}_{H^{1} (Ω)} + C (ℓ) {∥ U^{ε} (s) - U^{*} (s) ∥}_{H^{- 1} (Ω)}, \\ ∥ ζ^{ε} (s) - ζ^{*} {(s) ∥}_{L^{2} (\partial Ω)} \leq ℓ ∥ ζ^{ε} (s) - ζ^{*} {(s) ∥}_{H^{1} (\partial Ω)} + C (ℓ) {∥ ζ^{ε} (s) - ζ^{*} (s) ∥}_{H^{- 1} (\partial Ω)}, \end{matrix}

(61)

\forall ε > 0

and

\forall s \in [0, T]

, where

C (ℓ) \to 0

ℓ \to 0

Finally, relations (58)–(61) permit us to conclude that the assertion conducted in (42) holds true, ending the proof of Theorem 2.

Corollary 2.

Assume

U_{0} \in W_{\infty}^{2 - \frac{2}{p}} (Ω)

p_{_{2}} \frac{\partial}{\partial ν} U_{0} (x) - Δ_{Γ} U_{0} + p_{_{t}} U_{0} (x) = g_{_{f r}} (0, x)

\partial Ω

and

g_{_{f r}} \in W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)

. Then

U^{★} \in W_{_{Q}}

is a weak solution to the non-linear problem in (1).

Now we search the error of the numerical schemes (38) and (39) relative to

g_{_{d}}

and

g_{_{f r}}

. From Theorem 1 we know that

\forall g_{_{d}} \in L^{p} (Q)

and

g_{_{f r}} \in W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)

, the problem (8) has a unique solution

(U, ζ) \in W_{p}^{1, 2} (Q) \times W_{p}^{1, 2} (Σ)

. Moreover, (see (11))

\begin{matrix} {∥ U ∥}_{W_{p}^{1, 2} (Q)} + {∥ ζ ∥}_{W_{p}^{1, 2} (Σ)} \\ \leq C [1 + ∥ U_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Ω)}^{3 - \frac{2}{p}} + ∥ ζ_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Γ)}^{3 - \frac{2}{p}} + ∥ g_{_{d}} ∥_{L^{3 p - 2} (Q)}^{\frac{3 p - 2}{p}} + {∥ g_{_{f r}} ∥}_{W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)}], \end{matrix}

(62)

with a fixed

ζ_{0} \in W_{\infty}^{2 - \frac{2}{p}} (Γ)

and

U_{0} \in W_{\infty}^{2 - \frac{2}{p}} (Ω)

verifying

p_{_{2}} \frac{\partial}{\partial ν} U_{0} - Δ_{Γ} U_{0} + p_{_{t}} U_{0} = g_{_{f r}} (0, x)

. Thus, we have

Theorem 3.

Let

g_{_{d}} \in L^{p} (Q)

and

g_{_{f r}} \in W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)

. Let

g_{_{d}}^{k} \subset L^{p} (Q)

and

g_{_{f r}}^{k} \subset W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)

be two sequences such that

g_{_{d}}^{k} ⟶ g_{_{d}}

L^{p} (Q)

and

g_{_{f r}}^{k} ⟶ g_{_{f r}}

W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)

k ⟶ \infty

. Denoted by

(U_{m}, ζ_{m}) \subset W_{p}^{1, 2} (Q) \times W_{p}^{1, 2} (Σ)

and

(U_{m, k}, ζ_{m, k}) \subset W_{p}^{1, 2} (Q) \times W_{p}^{1, 2} (Σ)

, the approximating sequences are given in (38) and ((39), for

(g_{_{d}}, g_{_{f r}})

and

(g_{_{d}}^{k}, g_{_{f r}}^{k})

, respectively, with

U_{0} \in W_{\infty}^{2 - \frac{2}{p}} (Ω)

fixed. Then,

\begin{matrix} \underset{m ⟶ \infty}{lim sup} [∥ U_{m, k} {- U ∥}_{L^{2} (Q)} + {∥ ζ_{m, k} - ζ ∥}_{L^{2} (Σ)}] \\ \leq C e^{C T} m a x \{\underset{(t, x) \in Q}{m a x} | g_{_{d}}^{k} - g_{_{d}} |, \underset{(t, x) \in Σ}{m a x} | g_{_{f r}}^{k} - g_{_{f r}} |\} \end{matrix}

(63)

\forall k \geq 1

, where

C > 0

depends on

| Ω |

, T, n, p,

p_{_{1}}

p_{_{2}}

p_{_{t}}

p_{_{r}}

p_{_{s}}

∥ U_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Ω)}

∥ g_{_{d}} ∥_{L^{p} (Q)}

and

∥ g_{_{f r}} ∥_{W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)}

In particular,

\exists (U_{m, k}, ζ_{m, k})

, denoted by

(U_{m_{k}}, ζ_{m_{k}})

, such that

(U_{m_{k}}, ζ_{m_{k}}) ⟶ (U, ζ)

L^{p} (Q) \times L^{p} (Σ)

and in

Q \times Σ

k ⟶ \infty

Proof.

Owing to (62) we assume that

\begin{matrix} ∥ U_{k} ∥_{W_{p}^{1, 2} (Q)} + {∥ ζ_{k} ∥}_{W_{p}^{1, 2} (Σ)} \\ \leq C \{1 + ∥ U_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Ω)}^{3 - \frac{2}{p}} + ∥ ζ_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Γ)}^{3 - \frac{2}{p}} + ∥ g_{_{d}}^{k} ∥_{L^{3 p - 2} (Q)}^{\frac{3 p - 2}{p}} + {∥ g_{_{f r}}^{k} ∥}_{W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)}\} \\ \leq C \{1 + ∥ U_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Ω)}^{3 - \frac{2}{p}} + ∥ ζ_{0} ∥_{W_{\infty}^{2 - \frac{2}{p}} (Γ)}^{3 - \frac{2}{p}} + ∥ g_{_{d}} ∥_{L^{3 p - 2} (Q)}^{\frac{3 p - 2}{p}} + {∥ g_{_{f r}} ∥}_{W_{p}^{1 - \frac{1}{2 p}, 2 - \frac{1}{p}} (Σ)}\}, \end{matrix}

where

C > 0

is interpreted as

M_{4}

in (12). This ensures the applicability of (14) in Theorem 1 with

U_{0}^{1} = U_{0}^{2}

and

ζ_{0}^{1} = ζ_{0}^{2}

obtains

\begin{matrix} ∥ U_{k} {- U ∥}_{W_{p}^{1, 2} (Q)} + {∥ ζ_{k} - ζ ∥}_{W_{p}^{1, 2} (Σ)} \\ \leq C_{1} e^{C T} m a x \{\underset{(t, x) \in Q}{m a x} | g_{_{d}}^{k} - g_{_{d}} |, \underset{(t, x) \in Σ}{m a x} | g_{_{f r}}^{k} - g_{_{f r}} |\}, \forall k \geq 1, \end{matrix}

(64)

where

C_{1} > 0

. For

k \geq 1

, Theorem 2 gives

(U_{m, k} (s, \cdot), ζ_{m, k} (s, \cdot) ⟶ (U_{k} (s, \cdot), ζ_{k} (s, \cdot)) i n L^{2} (Ω) \times L^{2} (\partial Ω),

uniformly for

s \in [0, T]

, as

m ⟶ \infty

. In particular,

\forall k \geq 1

we have

(U_{m, k}, ζ_{m, k}) ⟶ (U_{k}, ζ_{k}), i n L^{2} (Q) \times L^{2} (Σ), a s m ⟶ \infty .

(65)

On the base of the relation in (64) and owing to (20), we obtain

\begin{matrix} ∥ U_{m, k} {- U ∥}_{L^{2} (Q)} + {∥ ζ_{m, k} - ζ ∥}_{L^{2} (Σ)} \\ \leq ∥ U_{m, k} - U_{k} ∥_{L^{2} (Q)} + ∥ ζ_{m, k} - ζ_{k} ∥_{L^{2} (Σ)} + ∥ U_{k} {- U ∥}_{L^{2} (Q)} + {∥ ζ_{k} - ζ ∥}_{L^{2} (Σ)} \\ \leq ∥ U_{m, k} - U_{k} ∥_{L^{2} (Q)} + {∥ ζ_{m, k} - ζ_{k} ∥}_{L^{2} (Σ)} \\ + C_{1} e^{C T} m a x \{\underset{(t, x) \in Q}{m a x} | g_{_{d}}^{k} - g_{_{d}} |, \underset{(t, x) \in Σ}{m a x} | g_{_{f r}}^{k} - g_{_{f r}} |\}, \forall m, k \geq 1 . \end{matrix}

Using (65) we can substitute the above inequality into the superior limit as

m ⟶ \infty

to prove that (63) is correct.

The last statement in Theorem 3 follows directly on from (63). □

The general frameworl of the numerical algorithm to compute the approximate solution to problem (1) via the fractional steps scheme may be demonstrated as follows:

Begin alg-frac_sec-ord_dbc

i = 0 \to U_{0}

from (39)

_{3}

;

For

i = 0

perform

M_{ε} - 1

Compute

z (ε, \cdot)

from (39);

U^{ε} (i ε, \cdot) = z (ε, \cdot)

;

ζ^{ε} (i ε, \cdot) = U^{ε} (i ε, \cdot)

;

Compute

(U^{ε} ((i + 1) ε, \cdot), ζ^{ε} ((i + 1) ε, \cdot))

solving the linear system (38);

End-for;

End.

5. Conclusions

The main problem addressed in this work concerns the non-linear second-order reaction–diffusion equation with its principal part in divergence form with inhomogeneous dynamic boundary conditions. Provided that the initial and boundary data meet the appropriate regularity and compatibility conditions, the well-posedness of a classical solution to the non-linear problem is proven in this new formulation (Theorem 1). Precisely, the Leray–Schauder principle and

L^{p}

theory of linear and quasi-linear parabolic equations, via Lemma 7.4 (see [1]), were applied to prove the qualitative properties of solution

(U (t, x), ζ (t, x))

. More precisely, we cannot directly apply the

L^{p}

theory to problem (1) (or (3)). Thus, this makes the result of Lemma 7.4 in Choban and Moroşanu [1] (p. 114) very important. Moreover, the a priori estimates were made in

L^{p} (Q)

and

L^{p} (Σ)

which permit the derivation of higher-order regularity properties, that is,

(U (t, x), ζ (t, x)) \in W_{p}^{1, 2} (Q) \times W_{p}^{1, 2} (Σ)

. Thus, the classical method of bootstrapping (see Moroşanu and Motreanu [20]) can be avoided.

Let us note that, due to the presence of the terms

K (t, x, U (t, x))

, the non-linear operator H (see (17)) does not represent the gradient of the energy functional. Therefore, the new proposed second-order non-linear problem cannot be obtained from the minimisation of any energy cost functional, i.e., (1) is not a variational PDE model.

Furthermore, an iterative fractional step-type scheme was introduced to approximate problem (8). The convergence and error estimates were established for the proposed numerical scheme and a conceptual numerical algorithm was formulated. In this regards, we want to underline the solutions dependence in Theorem 2 on the physical parameters, which could be useful in future investigations regarding error analysis and numerical simulations.

The qualitative results obtained here could be later used in quantitative approaches to the mathematical model (1) (or (3)) as well as in the study of distributed and/or non-linear optimal boundary control problems governed by such a non-linear problem.

Numerical implementation of the conceptual algorithm, alg-frac_sec-ord_dbc, as well as various simulations regarding the physical phenomena described by the non-linear parabolic problem (1) represent a matter for further investigation.

Author Contributions

Conceptualization, C.F. and C.M.; methodology, C.M.; validation, C.F. and C.M.; writing—original draft preparation, C.M.; writing—review and editing, C.F.; visualization, C.F.; funding acquisition, C.F. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Fetecău, C.; Moroşanu, C. Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions. Axioms 2023, 12, 406. https://doi.org/10.3390/axioms12040406

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Fetecău C, Moroşanu C. Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions. Axioms. 2023; 12(4):406. https://doi.org/10.3390/axioms12040406

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Fetecău, Constantin, and Costică Moroşanu. 2023. "Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions" Axioms 12, no. 4: 406. https://doi.org/10.3390/axioms12040406

APA Style

Fetecău, C., & Moroşanu, C. (2023). Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions. Axioms, 12(4), 406. https://doi.org/10.3390/axioms12040406

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Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions

Abstract

1. Introduction

2. Results—Theorem 1

3. Proof of the Main Result—Theorem 1

4. Approximating Scheme—Convergence and Error Estimate

Convergence of the Numerical Schemes (38) and (39)

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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