Solvability and spectral properties of the boundary value problem for degenerating higher order parabolic equation

doi:10.1016/j.amc.2015.06.131

Applied Mathematics and Computation

Volume 268, 1 October 2015, Pages 1282-1291

https://doi.org/10.1016/j.amc.2015.06.131 Get rights and content

Abstract

In the present paper we study the boundary value problem for degenerating higher order parabolic equation. A priori estimate for the solution of the boundary value problem is obtained. The existence of unique regular and strong solutions of the problem is proved. The adjoint problem is also investigated.

Introduction

Many problems of heat conductivity, chemical kinetics, burning and thermal explosion are described by the parabolic equations. Boundary value problems for non-degenerating parabolic equations are investigated in works of Friedman [1], Browder [2], Mikhaylov [3], Il’in et al. [4], Gorenflo and Mainardi [5], Luchko [6], [7] and others. Degenerating elliptic and parabolic equations are studied by Il’in [8], [9], [10]. In work of Baouendi and Grisvard [11] the problem of finding a solution of the equation $\frac{\partial^{2 k} u}{\partial x^{2 k}} + {(- 1)}^{k} x \frac{\partial u}{\partial t} = f$ in domain Ω ⊂ (a < x < b) × (0 < t < T), a < 0 < b, satisfying the boundary conditions $\frac{\partial^{l} u}{\partial x^{l}} = 0, l = 0, 1, \dots, k - 1,$ on $x = a, x = b$ and the initial conditions $u (x, 0) = u_{0} (x), 0 < x < b, u (x, T) = u_{T} (x), a < x < 0$ is considered, where k is positive integer. The authors proved that if $x^{\frac{1}{2}} u_{0} (x) \in L_{2} (0, b), | x^{\frac{1}{2}} | u_{T} (x) \in L_{2} (a, 0), f \in L_{2} ([0, T], H^{- k}),$ then there exists a unique solution $u (x, t) \in L_{2} ([0, T], \overset{0}{H^{k}})$

This equation is parabolic equation with changing direction of time. The boundary value problems for such equations were studied in Tersenov’s monography [12].

In the present article we consider the equation $L u = f (x, t),$ where $L u \equiv (t^{m} \frac{\partial^{2 k}}{\partial x^{2 k}} + {(- 1)}^{k} \frac{\partial}{\partial t}) u,$ m is a positive number and k ≥ 1 is fixed integer. Below in Section 2 for Eq. (1.1) we study the boundary value problem in the domain $Ω = {(x, t) : 0 < x < p, 0 < t < T}$ . An inequality which we use at the proof of a priori estimate in the norm of the space $W_{2, m}^{k, 0} (Ω)$ for the solution of the problem is proved. Existence of unique regular and strong solutions of the problem and its continuous dependence on f(x, t) is proved. It is established that the spectrum of the problem is empty set. It is shown that the problem is Volterra property. The adjoint problem is also investigated.

Section snippets

Statement of the problem

Problem 1

Find the solution u(x, t) of Eq. (1.1) in the domain Ω satisfying conditions $\begin{matrix} \frac{\partial^{2 l} u}{\partial x^{2 l}} (0, t) = \frac{\partial^{2 l} u}{\partial x^{2 l}} (p, t) = 0, l = 0, 1, \dots, k - 1, 0 \leq t \leq T, \end{matrix}$ $\begin{matrix} u (x, 0) = 0, 0 \leq x \leq p . \end{matrix}$ Let $V (Ω) = {u : u \in C_{x, t}^{2 k - 1, 0} (\bar{Ω}), \frac{\partial^{2 k} u}{\partial x^{2 k}}, \frac{\partial u}{\partial t} \in C (Ω) \cap L_{2} (Ω) a n d c o n d i t i o n s (2.1), a n d (2.2) a r e t r u e},$ $W (Ω) = {f : f \in C_{x, t}^{2 k, 0} (\bar{Ω}), \frac{\partial^{2 k + 1} u}{\partial x^{2 k + 1}} \in L_{2} (Ω), \frac{\partial^{2 l} f}{\partial x^{2 l}} (0, t) = \frac{\partial^{2 l} f}{\partial x^{2 l}} (p, t) = 0, l = 0, 1, \dots, k} .$

We define the operator L $L u \equiv (t^{m} \frac{\partial^{2 k}}{\partial x^{2 k}} + {(- 1)}^{k} \frac{\partial}{\partial t}) u$ mapping the set V(Ω) into C(Ω). The closure of the operator L in L₂(Ω) we denote by $\bar{L} .$

Definition 2.1

The function u(x, t) ∈ V(Ω) is called the regular

A priori estimate of the solution

The following holds

Lemma 3.1

For any n, 1 ≤ n ≤ k and any function u(x, t) ∈ V(Ω) the following inequality is true ${∥ t^{\frac{m}{2 k} n} \frac{\partial^{n} u}{\partial x^{n}} ∥}_{L_{2} (Ω)}^{2} \leq \frac{1}{2} ({∥ t^{\frac{m}{2 k} (n - 1)} \frac{\partial^{n - 1} u}{\partial x^{n - 1}} ∥}_{L_{2} (Ω)}^{2} + {∥ t^{\frac{m}{2 k} (n + 1)} \frac{\partial^{n + 1} u}{\partial x^{n + 1}} ∥}_{L_{2} (Ω)}^{2}) .$

Proof

According to the definition of the norm in L₂(Ω) we have ${∥ t^{\frac{m}{2 k} n} \frac{\partial^{n} u}{\partial x^{n}} ∥}_{L_{2} (Ω)}^{2} = \int_{0}^{T} [\int_{0}^{p} t^{\frac{m}{2 k} (n + 1)} \frac{\partial^{n} u}{\partial x^{n}} t^{\frac{m}{2 k} (n - 1)} \frac{\partial}{\partial x} (\frac{\partial^{n - 1} u}{\partial x^{n - 1}}) d x] d t .$ Integrating the interior integral by parts with respect to x and taking into account (2.1), and (2.2) we get ${∥ t^{\frac{m}{2 k} n} \frac{\partial^{n} u}{\partial x^{n}} ∥}_{L_{2} (Ω)}^{2} = \int_{0}^{T} \int_{0}^{p} (- t^{\frac{m}{2 k} (n + 1)} \frac{\partial^{n + 1} u}{\partial x^{n + 1}}) t^{\frac{m}{2 k} (n - 1)} \frac{\partial^{n - 1} u}{\partial x^{n - 1}} d x d t .$

The regular solvability of Problem 1

We search a regular solution of Problem 1 in the form of the Fourier series $u (x, t) = \sum_{n = 1}^{\infty} u_{n} (t) X_{n} (x)$ expanded in complete orthonormal system $X_{n} (x) = \sqrt{\frac{2}{p}} \sin λ_{n} x, λ_{n} = \frac{π n}{p}, n \in N$ in L₂(0, p).

It is clear that the function u(x, t) satisfies condition (2.1). Let f ∈ W(Ω). We expand the function f(x, t) into the Fourier series in functions X_n(x) $f (x, t) = \sum_{n = 1}^{\infty} f_{n} (t) X_{n} (x),$ where $f_{n} (t) = \int_{0}^{p} f (x, t) X_{n} (x) d x, n = 1, 2, \dots .$

Substituting (4.1) and (4.2) into Eq. (1.1) we obtain the following equation for unknown function u_n(t) $u_{n}^{'} (t) + λ$

The strong solvability

Substituting (4.5) into (4.1) we get $u (x, t) = \int_{0}^{p} \int_{0}^{t} K (x, t; ξ, τ) f (ξ, τ) d ξ d τ$ where $K (x, t; ξ, τ) = \sum_{n = 1}^{\infty} X_{n} (x) X_{n} (ξ) \exp {- λ_{n}^{2 k} \frac{t^{m + 1} - τ^{m + 1}}{m + 1}} d τ .$ The following is true:

Theorem 5.1

For any f ∈ L₂(Ω) the unique strong solution of Problem 1 exists, continuously depends on f(x, t), satisfies the estimation (3.4) and it can be represented in the form (5.1).

Proof

Let f be an arbitrary function in L₂(Ω). From relation $C_{0}^{\infty} (Ω) \subset W (Ω) \subset L_{2} (Ω)$ follows that W(Ω) is dense in L₂(Ω). Then there exists a sequence {f_i} ⊂ W(Ω), i ∈ N such that ${∥ f_{i} - f ∥}_{L}$

Spectrum of Problem 1

Definition 6.1

The spectrum of the problem is the set of eigenvalues of the operator $\bar{L}$ of Problem 1.

Definition 6.2

[15]. A problem is called the Volterra property if the inverse operator of the problem is Volterra property.

Theorem 6.1

The spectrum of Problem 1 is the empty set.

Proof

From (3.4) and (5.1) we conclude that it is the defined operator $L^{- 1}$ on W(Ω) which is the inverse of the operator L and acts from W(Ω) to V(Ω) by the rule $(L^{- 1} f) (x, t) = \int_{0}^{p} \int_{0}^{T} \bar{K} (x, t; ξ, τ) f (ξ, τ) d τ d ξ,$ where $\bar{K} (x, t; ξ, τ) = θ (t - τ) K (x, t; ξ, τ),$ $θ (t) = 1 for t > 0 and θ (t) = 0 for t \leq 0 .$

The adjoint problem

We consider the equation $L^{*} υ = g (x, t),$ where $L^{*} υ \equiv (t^{m} \frac{\partial^{2 k}}{\partial x^{2 k}} - {(- 1)}^{k} \frac{\partial}{\partial t}) υ .$

Problem 1^*

Find the solution $υ (x, t)$ of Eq. (7.1) in the domain Ω satisfying the conditions $\begin{matrix} \frac{\partial^{2 l} υ (0, t)}{\partial x^{2 l}} = \frac{\partial^{2 l} υ (p, t)}{\partial x^{2 l}} = 0, l = 0, 1, 2, \dots, k - 1; 0 \leq t \leq T \end{matrix}$ $\begin{matrix} υ (x, T) = 0, 0 \leq x \leq p . \end{matrix}$ We denote by $V^{*} (Ω) = {υ (x, t) : υ \in C_{x, t}^{2 k - 1, 0} (\bar{Ω}) \cap C_{x, t}^{2 k, 1} (Ω), \frac{\partial υ}{\partial t}, \frac{\partial^{2 k} υ}{\partial x^{2 k}} \in L_{2} (Ω),$ $and conditions (7.3), and (7.4) are true},$ $W^{*} (Ω) = {g (x, t) : g \in C_{x, t}^{2 k, 0} (\bar{Ω}), \frac{\partial^{2 k + 1} g}{\partial x^{2 k + 1}} \in L_{2} (Ω) \cap C (Ω),$ $\frac{\partial^{2 l} g (0, t)}{\partial x^{2 l}} = \frac{\partial^{2 l} g (p, t)}{\partial x^{2 l}} = 0, l = 0, 1, \dots, k} .$ We define the operator L^* mapping the domain V^*(Ω) into C(Ω) by (7.2). Let $\bar{L^{*}}$ be

Acknowledgments

The author is grateful to Professor Shavkat Alimov for his insightful suggestions.

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Solvability and spectral properties of the boundary value problem for degenerating higher order parabolic equation

Abstract

Introduction

Section snippets

Statement of the problem

A priori estimate of the solution

The regular solvability of Problem 1

The strong solvability

Spectrum of Problem 1

The adjoint problem

Acknowledgments

J. Comput. Appl. Math.

Comput. Math. Appl.

J. Math. Anal. Appl.

Partial Differential Equations of Parabolic Type

Linear parabolic differential equations of arbitrary order; general boundary – value problems for elliptic equations

Proc. Natl. Acad. Sci. USA

On potentials of parabolic equations

Dokl. Akad. Nauk SSSR

Linear second order equations of parabolic type

Usp. Mat. Nauk

On degenerating equations of elliptic and parabolic type

Usp. Mat. Nauk

Degenerating elliptic and parabolic equations, research reports of higher schools

Phis. Math.