Semilinear parabolic problems with nonlocal Dirichlet boundary conditions

Abstract

A semilinear parabolic problem with a nonlocal Dirichlet boundary condition is studied. This article presents a new and very easy implementable numerical algorithm for computations. This is based on a suitable linearization in time. The derived algorithm is implicit and it does not need any iteration process to get a solution with the nonlocal boundary condition. The stability analysis has been performed and the convergence of approximations towards a solution of the continuous problem is shown. The uniqueness of a solution is proved. Error estimates for the time discretization are derived.

Keywords:

• semilinear parabolic problem
• nonlocal Dirichlet boundary condition
• well-posedness
• error estimates

AMS Subject Classifications:

• 35K58
• 65M20

1. Introduction

Parabolic partial differential equations with nonlocal boundary conditions (BCs) arise in modelling of a wide range of important application areas such as thermoelasticity, heat conduction process, chemical diffusion, medicine science, etc. (cf. eg. 1–3).

Linear problems with a nonlocal BCs have been studied in the last few decades. The Dirichlet BC of the type (for x on the Dirichlet boundary) arises, e.g. in the in one-dimensional theory of linear thermoelasticity – (c.f. 4,5). The physical meaning of such a non-standard BC can be interpreted as follows: the value of a solution at the boundary is a weighted average of the solution inside the domain. The well-posedness of this linear problem has been discussed in 6. The proofs there are based on the fixed-point argument and on the smallness of the integral kernel K, namely

Monotonic properties of solutions for nonlinear parabolic problems along with nonlocal BCs were addressed in 7. The technique in 6,7 is based on the comparison principle for parabolic problems. Stronger conditions on K were needed in numerical studies - c.f. 8–11. The stability of various numerical algorithms has been addressed in 12. All the papers mentioned above needed in some way the smallness of the integral kernel. There are examples of problems which are solvable even if the integral kernel is large. The author in 13 raised a conjecture that the smallness of the integral kernel is only a technical (not crucial) condition, but up to now it remains an open question.

Solvability subject to a nonlocal Robin BC has been studied in 14–17. A numerical study based on monotonicity method has been performed in 18.

Let Ω ⊂ ℝ^d d ≥ 1 be a bounded domain with the Lipschitz continuous boundary Γ. In this article we study the solvability of the following problem: Find u such that (1) Remark that this problem is not an inverse problem in the classical sense because the forward operator is not defined. However, this study will serve to solve in the future the inverse source problem associated to (1).

The right-hand side (RHS) f is assumed to be a global Lipschitz continuous function. Let us note that f can change the sign, thus we cannot use technique based on the comparison principle. We can also consider the RHS of the form f(t, x, u(x, t), ∇u(x, t)) adopting the global Lipschitz continuity in all components. But for ease of explanation we will focus only on the most difficult term, namely on f(∇u). But our technique can be easily applied to the more general case.

The main goal of this article is to show the well-posedness of the problem (1) under certain reasonable conditions. First, we will prove the uniqueness of a solution (Theorem 2.1). For the solvability (cf. Theorem 4.2) we will use the semi-discretization in time (Rothe's method). Here we approximate the transient nonlinear and nonlocal problem by a sequence of standard linear Dirichlet boundary-value problems (BVPs). These are well-posed. We investigate their stability and finally prove the convergence of approximations to a solution in suitable function spaces. The solvability of (1) is proved using smallness of the integral kernel K.

2. Uniqueness

First, we denote by (w, z)_M the standard L₂-scalar product of the functions w and z on a measurable set M, i.e. and the corresponding norm The subscript will be suppressed if M = Ω.

As is usual in papers of this sort, C, ϵ and C_ϵ will denote generic positive constants depending only on a priori known quantities, where ϵ is sufficiently small and C_ϵ is large.

A variational formulation of (1) is given as follows.

Find u ∈ C([0, T ], L₂(Ω)) ∩ L_∞((0, T) H¹ (Ω)) obeying ∂_tu ∈ L₂((0, T) L₂(Ω)) such that (2)

Now, we derive two basic inequalities, which we will use in the proofs. Their derivation is based on the Cauchy inequality. (3) with , and (4) with . The following theorem proves the uniqueness of a solution to (1).

Theorem 2.1

Let f be Lipschitz continuous, K ∈ H⁰(Ω × Ω) and assume that and Then the variational problem (2) admits at most one solution.

Proof

Take two solutions u and v of (1) and set z = u − v. Then subtract the corresponding variational formulations (2) for both solutions from each other and put for any t ∈ [0, T ]. We integrate the result in time and we successively deduce that Fix an ϵ < 1 − ⋆. Gronwall's lemma implies that for all t ∈ [0, T ]. This yields u(x, t) − v(x, t) = z(x, t) = 0 almost everywhere in Ω × (0, T), which concludes the proof.▪

Remark 1

One can also consider a time-dependent integral kernel. The proof technique will remain the same using the following conditions: and We have considered only the time independent integral kernels for the ease of explanation.

3. Rothe's method

The proof of existence of a solution to (1) will be based on the Rothe method of time discretization.

We divide the time interval [0, T ] into n ∈ ℕ equidistant subintervals [t_i−1, t_i] for t_i = iτ, where . We introduce the following notation: for any function w.

We suggest the following time-discrete variational formulation: (5) Let us note that at each time step we have to solve a linear elliptic BVP (the RHS is taken from the previous time step) with known (from the previous time step) Dirichlet boundary condition.

Lemma 3.1

Let the assumptions of Theorem 2.1 be satisfied and u₀ ∈ H²(Ω). Then there exists a unique u_i ∈ H¹(Ω) for any i = 1, … , n solving (5).

Proof

Set δu₀ ≔ ∂_tu(0) = f(∇u₀) + Δu₀ ∈ L₂(Ω). Find u_i in the form with . Then for Using (3) and (4) we easily see that the RHS can be seen as a linear bounded functional on . Lax–Milgram lemma concludes the proof.▪

Next lemma addresses the stability of u_i.

Lemma 3.2

Let the assumptions of Lemma 3.1 be fulfilled. Then there exists C > 0 such that

Proof

Put into (5) and sum it up for i = 1, … , j. We get Using basic estimates (3) and (4) we derive in a standard way that Fixing an and applying Gronwall's lemma, we conclude the proof.▪

4. Existence of a solution

We want to prolong the u_i (defined for t = t_i) from the discrete time-points to the whole interval [0, T]. We do it in two different ways: as a piecewise linear function in time and as a step function in time Using this notation, we can rewrite (5) as follows: (6)

Lemma 4.1

Let the assumptions of Lemma 3.1 be fulfilled. There exists a subsequence of u_n, which is a Cauchy sequence in L₂((0, T), H¹(Ω)).

Proof

Lemma 3.2 together with 19, Lemma 1.3.13] imply the existence of a subsequence of u_n (denoted, by the same symbol again) which is a Cauchy sequence in C([0, T ], L₂(Ω)). Moreover, in H¹(Ω) for all t ∈ [0, T ] and ∂_tu_n ⇀ ∂_tu in L₂(0, T), L₂(Ω)).

Subtract (6) for n = s from (6) for n = r. Then put and integrate over (0, T). One obtains Now, we separately estimate all the terms R₁, … , R₄. The procedure is standard. We employ the Cauchy inequality, (3), (4) and Lemma 3.2. We may write and Similarly as for R₃ we also estimate R₄ as According to the fact that u_n is a Cauchy sequence in C([0, T ], L₂(Ω)), we deduce that which concludes the proof.▪

Now, we are in a position to prove the solvability of (1).

Theorem 4.2

Let the assumptions of Lemma 3.1 be fulfilled. Then the direct problem (1) admits a solution u ∈ C([0, T ], L₂(Ω) ∩ L_∞((0, T), H¹(Ω)) obeying ∂_tu ∈ L₂((0, T), L₂(Ω)).

Proof

Integrate (6) over (0, t) and get (7) Use Pass to n → ∞ in (7) to get Differentiating with respect to t implies that u is a weak solution to (1). We have to check the behaviour of u_n on the boundary Γ. We may write Thus Using the Cauchy inequality we see that (for t ∈ [t_i−1, t_i]) Analogously we deduce that and It holds that Hence Using the well-known inequality 20 (8) we obtain Passing to the limit when τ → 0, we get which is valid for all small ϵ > 0. Therefore Having this, we see that the u solves (1).▪

The convergence in Lemma 4.1 is valid for a subsequence. The fact that the problem (1) admits a unique solution implies that the convergence in Lemma 4.1 is valid for the whole sequence.

5. Error estimates

An important assumption for the solvability of (1) was the Lipschitz continuity of f and u₀ ∈ H²(Ω). If we would like to get error estimates for the time discretization, we need better a priori estimates. To get these we will differentiate (1) in time. For the time-discrete case it means that the initial datum u₀ and also f have to be smoother. Namely, if we define (cf. Lemma 3.1) we see that We need a similar form to (5) for i = 0. We can write (9) If u₀ ∈ H³(Ω) and f is Lipschitz continuous, then δu₀, u₋₁ ∈ H¹(Ω). Let us define We see that if f ∈ C² and u₀ ∈ H⁴(Ω) ∩ H^2,4(Ω), then δu₋₁ ∈ L₂(Ω).

Lemma 5.1

Let the assumptions of Lemma 3.1 be fulfilled. Moreover, assume f ∈ C² and u₀ ∈ H⁴(Ω) ∩ H^2,4(Ω). Then

Proof

Subtract (5) for i = i − 1 from (5). For i = 0 use (9) instead of (5). Put and sum it up for i = 1, … , j. We get The last term on the RHS can be estimated as follows: The rest of the proof follows exactly the same line as the one in Lemma 3.2 and therefore we omit it.▪

The following theorem derives the error estimates for the time discretization.

Theorem 5.2

Let the assumptions of Lemma 5.1 be fulfilled. Then and

Proof

Subtract (6) from (2) and set . Then we integrate it in time over (0, ξ). We may write The left-hand side can be rewritten as The terms on the RHS can be estimated in a standard way as follows: and Putting things together and choosing ϵ < 1 − ⋆ we get which is valid for all ξ ∈ [0, T]. Applying Gronwall's argument, we get Using this and according to the trace theorem we observe that Further we may write and which concludes the proof.▪

6. Conclusions

In this work, we studied the well-posedness of the identification problem (1). We showed the uniqueness and the existence of a solution in appropriate function spaces. Besides this we also designed a numerical scheme for computations and derived the error estimates for the suggested algorithm. This study will serve to solve the inverse source problem associated to (1) in the future.

Acknowledgements

This work was supported by the BOF/GOA-project no. 01G006B7 at Ghent University.