Elliptic and Parabolic Equations with Involution and Degeneration at Higher Derivatives

Abstract

We study the solvability in Sobolev spaces of boundary value problems for elliptic and parabolic equations with variable coefficients in the presence of an involution (involutive deviation) at higher derivatives, both in the nondegenerate and degenerate cases. For the problems under study, we prove the existence theorems as well as the uniqueness of regular solutions, i.e., those that have all weak derivatives in the equation.

1. Introduction

A function $\phi \left(x\right)$ on some subset of the reals is called an involution on E if $\phi \left(\phi \right(x\left)\right)=x$ for $x\in E$. The simplest example of an involution in the case of $E=[0,1]$ is a linear-fractional function

$$\phi \left(x\right)=\frac{b(1-x)}{cx+b},b\in \mathbb{R},c\in \mathbb{R},b(b+c)\ne 0$$

(for $c=0$, a linear function); other examples can be found in [1].

This article is devoted to the study of the solvability of boundary value problems for second-order differential equations in which certain derivatives of the unknown solution are given not only at the current points but also at the points determined by some involution. Similar equations (elliptic, parabolic, and hyperbolic) have been actively studied recently (see [2,3,4,5,6,7,8,9,10]) but, in these works, all the equations under consideration either have a special form or do not contain an involution in the higher part. Moreover, the involution therein is the simplest; i.e., linear.

Differential equations with involution in this paper differ essentially from the equations studied by our predecessors. Firstly, in the present article, we consider equations with variable coefficients. Secondly, the equations contain an involution (involutive deviation) of a general type in the higher part. Finally, we study equations with degeneration (note that boundary value problems for differential equations with involution and degeneration were not studied earlier).

The direct object of research in the present article is boundary value problems for elliptic and parabolic equations with variable coefficients and general involution at higher derivatives. The method for studying the solvability of these problems differs significantly from the methods of the predecessors.

All constructions and arguments in the article are carried out on the basis of the Lebesgue spaces $L_p$ and the Sobolev spaces $W_p^l$. The necessary definitions and descriptions of these spaces can be found in [11,12,13].

Let us clarify that the purpose of this paper is to prove the existence and uniqueness of regular solutions to a boundary value problem, i.e., solutions that have all weak derivatives in the corresponding equation.

The article consists of four parts. In the first part, we give the statements of the problems under study and their equivalent formulations. In the second part, we study the solvability of the problems posed in the first part in the nondegenerate case. The third part of the paper is devoted to the solvability of boundary value problems for elliptic and parabolic equations with involution in the degenerate case. Finally, in the fourth part of the article, some generalizations and enhancements of the results of the second and third parts are described.

2. Statement of the Problems

Let $\mathsf{\Omega }$ be the interval $(0,1)$ of $Ox$, and let Q be a rectangle $\mathsf{\Omega }\times (0,T)$ of a finite height T. Next, suppose that $\phi \left(x\right)$ is a given involution of $[0,1]$, while $a(x,t)$, $c(x,t)$ and $f(x,t)$ are functions on $\overline{Q}$.

Boundary Value Problem I.Find a function $u(x,t)$ that is a solution in Q to the equation

$$u_t(x,t)-u_{xx}(x,t)-a(x,t)u_{xx}(\phi \left(x\right),t)+c(x,t)u(x,t)=f(x,t)$$

(1)

and satisfies the conditions

$$u(0,t)=u(1,t)=0,t\in (0,T),$$

(2)

$$u(x,0)=0,x\in \mathsf{\Omega }.$$

(3)

Boundary Value Problem II.Find a function $u(x,t)$ solving the following equation in Q

$$u_{tt}(x,t)+u_{xx}(x,t)+a(x,t)u_{xx}(\phi \left(x\right),t)-c(x,t)u(x,t)=f(x,t)$$

(4)

satisfying (2) and (3) such that

$$u(x,T)=0,x\in \mathsf{\Omega }.$$

(5)

In case $a(x,t)\equiv 0$, Boundary Value Problems I and II are the well-studied conventional boundary value problems for parabolic and elliptic equations (i.e., for the equations without involution). If $a(x,t)$ is not identically zero, then Problems I and II were studied earlier only in the case $a(x,t)\equiv const$ and $\phi \left(x\right)=1-x$.

Put

$$a_1(x,t)=1-a(x,t)a(\phi \left(x\right),t),$$

$$c_1(x,t)=a(x,t)c(\phi \left(x\right),t),$$

$$f_1(x,t)=f(x,t)-a(x,t)f(\phi \left(x\right),t).$$

Suppose that

$$a_1(x,t)\ge {\overline{a}}_1>0\mathrm{for}(x,t)\in \overline{Q},{\overline{a}}_1=const.$$

(6)

Replace x by $\phi \left(x\right)$ in (1). We obtain

$$u_t(\phi \left(x\right),t)-u_{xx}(\phi \left(x\right),t)-a(\phi \left(x\right),t)u_{xx}(x,t)+c(\phi \left(x\right),t)u(\phi \left(x\right),t)=f(\phi \left(x\right),t)$$

(7)

Equalities (1) and (7) are a linear system for the functions $u_{xx}(x,t)$ and $u_{xx}(\phi \left(x\right),t)$. In view of condition (6), this system is uniquely solvable. Finding this solution, we obtain

$$u_t(x,t)-a(x,t)u_t(\phi \left(x\right),t)-a_1(x,t)u_{xx}(x,t)+c(x,t)u(x,t)-c_1(x,t)u(\phi \left(x\right),t)=f_1(x,t).$$

(8)

Now, replace x by $\phi \left(x\right)$ in (4). We arrive at the equality which, together with (4), again gives a linear system for $u_{xx}(x,t)$ and $u_{xx}(\phi \left(x\right),t)$. Again, by condition (6), the so-constructed system has a unique solution, and this solution yields

$$u_{tt}(x,t)-a(x,t)u_{tt}(\phi \left(x\right),t)+a_1(x,t)u_{xx}(x,t)-c(x,t)u(x,t)+c_1(x,t)u(\phi \left(x\right),t)=f_1(x,t).$$

(9)

Equalities (8) and (9) enable us to pass from the boundary of Boundary Value Problems I and II to the two new problems:

Boundary Value Problem I${}^{\prime }$.Find a function $u(x,t)$ solving (8) in Q and satisfying (2) and (3).

Boundary Value Problem II${}^{\prime }$.Find a function $u(x,t)$ solving (9) in Q and satisfying (2), (3), and (5).

Under condition (6), Boundary Value Problems I and I${}^{\prime }$ and II and II${}^{\prime }$ are equivalent; if (6) fails, then Problems I and I${}^{\prime }$ and II and II${}^{\prime }$ can be regarded as independent problems for differential equations (8) and (9).

Note that, under condition (6), Equations (8) and (9) lack degeneration; if we replace (6) by the condition

$$a_1(x,t)\ge 0\mathrm{for}(x,t)\in \overline{Q},$$

(10)

then (8) and (9) become a degenerate equation with involution.

Conditions (6) and (10) determine in this article whether Boundary Value Problems I and II have or lack degeneration.

Observe also the following: Define the operators A and C in Q:

$$\left(Av\right)(x,t)=v(x,t)-a(x,t)v\left(\phi \right(x),t),$$

$$\left(Cv\right)(x,t)=c(x,t)v(x,t)-c_1(x,t)v(\phi \left(x\right),t).$$

Using these operators, (8) and (9) are represented as equations not solved for the derivative (see [14,15]); namely,

$$A{u}_t-a_1{u}_{xx}+Cu=f_1(x,t),$$

$$A{u}_{tt}-a_1{u}_{xx}-Cu=f_1(x,t).$$

Here, A, regarded as an operator from $L_2\left(Q\right)$ to $L_2\left(Q\right)$, can be either invertible or uninvertible.

3. Solvability of Boundary Value Problems I and II in the Nondegenerate Case

The existence and uniqueness of regular solutions to Boundary Value Problems I and II in the nondegenerate case is established by the method of continuation in a parameter [16] (Chapter III, §14) and a priori estimates.

Theorem 1.

Suppose that

$$a(x,t)\in C^1\left(\overline{Q}\right),c(x,t)\in C\left(\overline{Q}\right),$$

(11)

$$\phi \left(x\right)\in C^1\left(\overline{\mathsf{\Omega }}\right),-{\phi }_1\le {\phi }^{\prime }\left(x\right)\le -{\phi }_0<0\mathit{for}x\in \overline{\mathsf{\Omega }},$$

(12)

$$\sqrt{{\phi }_1}\underset{\overline{Q}}{max}\left|a(x,t)\right|<1$$

(13)

and assume condition (6). Then, for every $f(x,t)\in L_2\left(Q\right)$, Boundary Value Problem I has a solution $u(x,t)\in W_2^{2,1}\left(Q\right)$, which is unique.

Proof. 

Let $\lambda \in [0,1]$. Consider the family of boundary value problems: Find a solution $u(x,t)$ satisfying in Q the equation

$$u_t(x,t)-a_1(x,t)u_{xx}(x,t)+c(x,t)u(x,t)-\lambda \left[a(x,t)u_t(\phi \left(x\right),t)+c_1(x,t)u(\phi \left(x\right),t)\right]=f_1(x,t)$$

(14)

as well as conditions (2) and (3). By the theorem on continuation in a parameter, this problem is solvable in $W_2^{2,1}\left(Q\right)$ if (1) $f_1(x,t)\in L_2\left(Q\right)$; (2) the boundary value problems (14), (2), and (3) are solvable in $W_2^{2,1}\left(Q\right)$ for $\lambda =0$, $f_1(x,t)\in L_2\left(Q\right)$; (3) all possible solutions $u(x,t)$ to the boundary value problems (2), (3) and (14), satisfy the a priori estimate

$${\left|\right|u\left|\right|}_{W_2^{2,1}\left(Q\right)}\le R_0\left|\right|f_1{\left|\right|}_{L_2\left(Q\right)}$$

(15)

with a constant $R_0$ defined only by $a(x,t)$, $c(x,t)$, $\phi \left(x\right)$, and T.

The membership $f_1(x,t)\in L_2\left(Q\right)$ is obvious. Next, the solvability of problems (2), (3) and (14) in $W_2^{2,1}\left(Q\right)$ for $\lambda =0$ and under conditions (6) and (11) is well-known (see [17]). Show that all possible solutions $u(x,t)$ to the boundary value problems (2), (3) and (14) satisfy (15) uniformly in $\lambda$.

Write (14) in the variables $(x,\tau )$, multiply the result by $u_{\tau }(x,\tau )$, and integrate over the rectangle $\mathsf{\Omega }\times (0,t)$. We obtain

$$\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{u}_{\tau }^2(x,\tau )dxd\tau +\frac{1}{2}\underset{\mathsf{\Omega }}{\int }{a}_1(x,t)u_x^2(x,t)dx=$$

$$=\lambda \underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }a(x,\tau )u_{\tau }(\phi \left(x\right),\tau )u_{\tau }(x,\tau )dxd\tau +\frac{1}{2}\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{a}_{1\tau }(x,\tau )u_x^2(x,\tau )dxd\tau -$$

$$-\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{a}_{1x}(x,\tau )u_x(x,\tau )u_{\tau }(x,\tau )dxd\tau -\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }c(x,\tau )u(x,\tau )u_{\tau }(x,\tau )dxd\tau +$$

$$+\lambda \underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{c}_1(x,\tau )u(\phi \left(x\right),\tau )u_{\tau }(x,\tau )dxd\tau +\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{f}_1(x,\tau )u_{\tau }(x,\tau )dxd\tau ,$$

(16)

Estimate the first summand on the right-hand side of (16):

$$\left|\lambda \underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }a(x,\tau )u_{\tau }(\phi \left(x\right),\tau )u_{\tau }(x,\tau )dxd\tau \right|\le$$

$$\le \underset{\overline{Q}}{max}\left|a(x,\tau )\right|{\left(\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{u}_{\tau }^2(x,\tau )dxd\tau \right)}^{\frac{1}{2}}{\left(\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{u}_{\tau }^2(\phi \left(x\right),\tau )dxd\tau \right)}^{\frac{1}{2}}=$$

$$=\underset{\overline{Q}}{max}\left|a(x,\tau )\right|{\left(\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{u}_{\tau }^2(x,\tau )dxd\tau \right)}^{\frac{1}{2}}{\left(\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }\left|{\phi }^{\prime }\left(x\right)\right|u_{\tau }^2(x,\tau )dxd\tau \right)}^{\frac{1}{2}}\le$$

$$\le \sqrt{{\phi }_1}\underset{\overline{Q}}{max}\left|a(x,\tau )\right|\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{u}_{\tau }^2(x,\tau )dxd\tau .$$

Using this equality and (13), and applying Young’s inequality, making the change $y=\phi \left(x\right)$ in the penultimate integral on the right-hand side of (16), recalling the inequality

$$\underset{\mathsf{\Omega }}{\int }{u}^2(x,t)dx\le T\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{u}_{\tau }^2(x,\tau )dxd\tau ,$$

and, finally, applying Gronwall’s lemma, we obtain that the solutions $u(x,t)$ to the boundary value problem (2), (3) and (14) satisfy the a priori estimate

$$\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{u}_{\tau }^2(x,\tau )dxd\tau +\underset{\mathsf{\Omega }}{\int }{u}_x^2(x,t)dx\le R_1\left|\right|f_1{\left|\right|}_{L_2\left(Q\right)}^2,$$

(17)

where the constant $R_1$ is defined only by $a(x,t)$, $c(x,t)$, $\phi \left(x\right)$, and T.

At the next step, multiply (14) by $u_{xx}(x,t)$ and integrate the result over Q. Using condition (6) and inequality (17), it is not hard to obtain the second a priori estimate of solutions $u(x,t)$ to problems (2), (3) and (14):

$$\underset{Q}{\int }{u}_{xx}^2(x,t)dxdt\le R_2\left|\right|f_1{\left|\right|}_{L_2\left(Q\right)}^2,$$

(18)

with the constant $R_2$ defined only by $a(x,t)$, $c(x,t)$, $\phi \left(x\right)$, and T.

Estimates (17) and (18) imply the desired estimate (15).

The above arguments mean that the boundary value problems (2), (3) and (14) are solvable in $W_2^{2,1}\left(Q\right)$ for all $\lambda$ in $[0,1]$; in particular, for $\lambda =1$. However, then Boundary Value Problem I is solvable in this space.

The uniqueness of a solution to Boundary Value Problem I in $W_2^{2,1}\left(Q\right)$ is obvious.

The theorem is completely proved. □

The proof of the solvability of Boundary Value Problem II in the nondegenerate case differs from the proof of Theorem 1 only by the circumstance that it is impossible to apply Gronwall’s lemma to elliptic equations.

Theorem 2.

Suppose that

$$a(x,t)\in C\left(\overline{Q}\right),c(x,t)\in C^2\left(\overline{Q}\right),$$

(19)

$$\phi \left(x\right)\in C^1\left(\overline{\mathsf{\Omega }}\right),-{\phi }_1\le {\phi }^{\prime }\left(x\right)\le -{\phi }_0\mathit{for}x\in \overline{\mathsf{\Omega }},$$

(20)

$$c(x,t)\ge 0,c_{xx}(x,t)\le 0\mathit{for}(x,t)\in \overline{Q}$$

(21)

on assuming conditions (6) and (13). Then, for every $f(x,t)\in L_2\left(Q\right)$, Boundary Value Problem II has a solution in $W_2^2\left(Q\right)$, and this solution is unique.

Proof. 

We again use the method of continuation in a parameter. Namely, for $\lambda \in [0,1]$, consider the problem: Find a function $u(x,t)$ satisfying in Q the equation

$$u_{tt}(x,t)+u_{xx}(x,t)+\lambda a(x,t)u_{xx}(\phi \left(x\right),t)-c(x,t)u(x,t)=f(x,t)$$

(22)

as well as conditions (2), (3), and (5). For $\lambda =0$, this problem is solvable in $W_2^2\left(Q\right)$ [12]; its solvability for all $\lambda$ follows from the a priori estimate

$${\left|\right|u\left|\right|}_{W_2^2\left(Q\right)}\le R_0^{\prime }{\left|\right|f\left|\right|}_{L_2\left(Q\right)}$$

(23)

of all possible solutions $u(x,t)$ to problems (2), (3), (5) and (22).

Show that the desired estimate holds.

Consider the equality

$$\underset{Q}{\int }\left[u_{tt}(x,t)+u_{xx}(x,t)+\lambda a(x,t)u_{xx}(\phi \left(x\right),t)-c(x,t)u(x,t)\right]u_{xx}(x,t)dxdt=$$

$$=\underset{Q}{\int }f(x,t)u_{xx}(x,t)dxdt.$$

Integrating by parts and estimating the integral of the summand with involution by Hölder’s inequality, the change $y=\phi \left(x\right)$ and conditions (20) and (13). Moreover, using (21), we obtain the first a priori estimate

$$\underset{Q}{\int }\left(u_{xt}^2+u_{xx}^2\right)dxdt\le R_1^{\prime }\underset{Q}{\int }{f}^{2}dxdt.$$

(24)

Subsequent estimates are obviously deduced from (24). Summarizing, this yields the desired estimate (23).

As we already said above, estimate (23) and the theorem on continuation in a parameter imply the solvability of Boundary Value Problem II in $W_2^2\left(Q\right)$.

The uniqueness of a solution to Boundary Value Problem II in $W_2^2\left(Q\right)$ is obvious.

The theorem is proved. □

4. Solvability of Boundary Value Problems I and II in the Degenerate Case

Suppose that, in (1) and (4), a is a function only of t and assume that the condition

$$\left|a\right(t\left)\right|\le 1\mathrm{for}t\in [0,T]$$

(25)

is fulfilled instead of (6).

This condition means that equations (8) and (9) are degenerate. Moreover, even in the simplest case of a linear involution under condition (25), conditions (6) and (13) are not fulfilled. Consequently, some additional conditions are necessary for the existence of regular solutions to Boundary Value Problems I and II.

Theorem 3.

Suppose (25), and also the conditions

$$a\left(t\right)\in C^1\left([0,T]\right),c(x,t)\in C^1\left(\overline{Q}\right),$$

(26)

$$\phi \left(x\right)\in C^1\left(\overline{\mathsf{\Omega }}\right),-{\phi }_1\le {\phi }^{\prime }\left(x\right)\le -{\phi }_0\mathit{for}x\in \overline{\mathsf{\Omega }},$$

(27)

$$\sqrt{{\phi }_1}\underset{[0,T]}{max}\left|a\left(t\right)\right|\le 1,{\phi }_1\underset{[0,T]}{max}\left|a\left(t\right)\right|\le 1.$$

(28)

Then, for every $f(x,t)\in L_2\left(0,T;W_2^2\left(\mathsf{\Omega }\right)\bigcap {\mathring{W}}_2^1\left(\mathsf{\Omega }\right)\right)$, Boundary Value Problem I has a solution $u(x,t)$ such that $u(x,t)\in L_{\infty }\left(0,T;W_2^2\left(\mathsf{\Omega }\right)\bigcap {\mathring{W}}_2^1\left(\mathsf{\Omega }\right)\right)$, $u_t(x,t)\in L_2\left(Q\right)$.

Proof. 

Proceed by the regularization method. Let $\epsilon$ be a positive real. Consider the boundary value problem: Find a function $u(x,t)$ satisfying in Q the equation

$$u_t(x,t)-(1+\epsilon )u_{xx}(x,t)-a\left(t\right)u_{xx}(\phi \left(x\right),t)+\epsilon u_{xxxx}(x,t)+c(x,t)u(x,t)=f(x,t)$$

(29)

as well as (2), (3), and the condition

$$u_{xx}(0,t)=u_{xx}(1,t)=0,t\in (0,T).$$

(30)

Here, (29) is a parabolic equation without degeneration; the solvability of problems (2), (3), (29) and (30) in $W_2^{4,1}\left(Q\right)$ for $\epsilon$ fixed and $f(x,t)\in L_2\left(Q\right)$ is not hard to prove by the method of continuation in a parameter and the a priori estimates obtained by the scheme of proving Theorem 1.

Show that, under the hypotheses of the theorem, solutions $u(x,t)$ to problems (2), (3), (29) and (30) admit a priori estimates uniform in $\epsilon$, which make it possible to organize passage to the limit.

Consider the equality

$$\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }\left\{u_t(x,\tau )-(1+\epsilon )u_{xx}(x,\tau )-a\left(\tau \right)u_{xx}(\phi \left(x\right),\tau )+\epsilon u_{xxxx}(x,\tau )+\right.$$

$$\left.+c(x,\tau )u(x,\tau )\right\}{u}_{xx}(x,\tau )dxd\tau =\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }f(x,\tau )u_{xx}(x,\tau )dxd\tau .$$

Integrating by parts both on the left- and right-hand sides of this equality, making the change $y=\phi \left(x\right)$ in the integral with involution, using conditions (26)–(28), and, finally, applying Gronwall’s lemma, we conclude that solutions $u(x,t)$ to problems (2), (3), (29) and (30) satisfy the estimate

$$\underset{\mathsf{\Omega }}{\int }{u}_t^2(x,t)dx+\epsilon \underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{u}_{xxx}^2(x,\tau )dxd\tau \le R_3\underset{Q}{\int }\left(f^2+f_x^2\right)dxdt$$

(31)

with a constant $R_3$ defined only by $a\left(t\right)$, $c(x,t)$, $\phi \left(x\right)$, and T.

At the next step, consider the equality

$$\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }\left\{u_t(x,\tau )-(1+\epsilon )u_{xx}(x,\tau )-a\left(\tau \right)u_{xx}(\phi \left(x\right),\tau )+\epsilon u_{xxxx}(x,\tau )+\right.$$

$$\left.+c(x,\tau )u(x,\tau )\right\}{u}_{xxxx}(x,\tau )dxd\tau =\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }f(x,\tau )u_{xxxx}(x,\tau )dxd\tau .$$

Integrating by parts once again, making the change $y=\phi \left(x\right)$, using conditions (26)–(28), and applying Gronwall’s lemma, we conclude that solutions $u(x,t)$ to problems (2), (3), (29) and (30) satisfy the estimate

$$\underset{\mathsf{\Omega }}{\int }{u}_{xx}^2(x,t)dx+\epsilon \underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{u}_{xxxx}^2(x,\tau )dxd\tau \le R_4\underset{Q}{\int }\left(f^2+f_x^2+f_{xx}^2\right)dxdt$$

(32)

with a constant $R_4$ defined only by $a\left(t\right)$, $c(x,t)$, $\phi \left(x\right)$, and T.

The next estimate

$$\underset{0}{\overset{t}{\int }}\underset{\mathsf{\Omega }}{\int }{u}_t^2(x,\tau )dxd\tau \le R_5\underset{Q}{\int }\left(f^2+f_x^2+f_{xx}^2\right)dxdt$$

(33)

is obviously from the previous estimates; here, the constant $R_5$ in this estimate is defined only by $a\left(t\right)$, $c(x,t)$, $\phi \left(x\right)$ and T.

Estimates (31)–(33) are sufficient for passing to the limit.

Choose a sequence ${\left\{{\epsilon }_m\right\}}_{m=1}^{\infty }$ of positive reals such that ${\epsilon }_m\to 0$ as $m\to \infty$. Next, let ${\left\{u_m(x,t)\right\}}_{m=1}^{\infty }$ be a sequence of solutions to problems (2), (3), (29) and (30) corresponding to ${\epsilon }_m$. The family ${\left\{u_m(x,t)\right\}}_{m=1}^{\infty }$ satisfies the a priori estimates (31)–(33). These estimates and the reflexivity of a Hilbert space mean that there exists a sequence ${\left\{m_k\right\}}_{k=1}^{\infty }$ of naturals and a function $u(x,t)$, such that as $k\to \infty$, we have

$$u_{m_k}(x,t)\to u(x,t)\mathit{weakly}\mathit{in}{W}_2^{2,1}\left(Q\right),$$

$${\epsilon }_{m_k}{u}_{m_{k}xxxx}(x,t)\to 0\mathit{weakly}\mathit{in}{L}_2\left(Q\right),$$

$$u_{m_{k}xx}(\phi \left(x\right),t)\to u_{xx}(\phi \left(x\right),t)\mathit{weakly}\mathit{in}{L}_2\left(Q\right).$$

Obviously, the limit function $u(x,t)$ is a desired solution to Boundary Value Problem I.

The theorem is proved. □

The study of the solvability of Boundary value Problem II is in general carried out by the same scheme as for Boundary Value Problem I; the only difference is that it is impossible to use Gronwall’s lemma in the elliptic case.

Theorem 4.

Suppose (25), as well as the conditions

$$a\left(t\right)\in C^2\left([0,T]\right),c(x,t)\in C^2\left(\overline{Q}\right),$$

$$\phi \left(x\right)\in C^1\left(\overline{\mathsf{\Omega }}\right),-{\phi }_1\le {\phi }^{\prime }\left(x\right)\le -{\phi }_0<0\mathit{for}x\in \overline{\mathsf{\Omega }},$$

$$\sqrt{{\phi }_1}\underset{[0,T]}{max}\left|a\left(t\right)\right|\le 1,{\phi }_1\underset{[0,T]}{max}\left|a\left(t\right)\right|\le 1,$$

$$c(x,t)\ge 0,c_{xx}(x,t)\le 0\mathit{for}(x,t)\in \overline{Q}.$$

Then, for every $f(x,t)in{L}_2\left(0,T;W_2^2\left(\mathsf{\Omega }\right)\bigcap {\mathring{W}}_2^1\left(\mathsf{\Omega }\right)\right)$, Boundary Value Problem II has a solution $u(x,t)$ such that $u(x,t)\in L_{\infty }\left(0,T;W_2^2\left(\mathsf{\Omega }\right)\bigcap {\mathring{W}}_2^1\left(\mathsf{\Omega }\right)\right)$, $u_t(x,t)\in L_2\left(0,T;W_2^2\left(\mathsf{\Omega }\right)\bigcap {\mathring{W}}_2^1\left(\mathsf{\Omega }\right)\right)$, $u_{tt}(x,t)\in L_2\left(Q\right)$.

Proof. 

Proceed by the regularization method once again. For a positive real $\epsilon$, consider the boundary value problem: Find a function $u(x,t)$ satisfying in Q the equation

$$u_{tt}(x,t)+(1+\epsilon )u_{xx}(x,t)+a\left(t\right)u_{xx}(\phi \left(x\right),t)-\epsilon u_{xxxx}(x,t)-c(x,t)u(x,t)=f(x,t)$$

(34)

as well as (2), (3), (5), and (30). For $\epsilon$ fixed and $f(x,t)\in L_2\left(Q\right)$, the problem has a solution $u(x,t)\in W_2^{4,2}\left(Q\right)$ (this is proved by the method of continuation in a parameter). Furthermore, solutions $u(x,t)$ to problems (2), (3), (5), (30) and (34) satisfy the a priori estimates

$$\underset{Q}{\int }{u}_{xt}^{2}dxdt+\epsilon \underset{Q}{\int }{u}_{xxx}^{2}dxdt\le R_6\underset{Q}{\int }\left(f^2+f_x^2\right)dxdt,$$

(35)

$$\underset{Q}{\int }{u}_{xxt}^{2}dxdt+\epsilon \underset{Q}{\int }{u}_{xxxx}^{2}dxdt\le R_7\underset{Q}{\int }\left(f^2+f_x^2+f_{xx}^2\right)dxdt,$$

(36)

$$\underset{Q}{\int }{u}_{tt}^{2}dxdt\le R_8\underset{Q}{\int }\left(f^2+f_x^2+f_{xx}^2\right)dxdt,$$

(37)

where the constants $R_6$, $R_7$, and $R_8$ are defined only by $\phi \left(x\right)$, $a\left(t\right)$, and $c(x,t)$; these estimates are deduced by analyzing the equalities that result after multiplying (34) by $u_{xx}(x,t)$, $-u_{xxxx}(x,t)$, and $u_{tt}(x,t)$ and subsequently integrating over Q. These estimates and the reflexivity of a Hilbert space make it possible to appropriately choose a sequence $\left\{u_{m_k}(x,t)\right\}$ of solutions to problems (2), (3), (5) and (34) with $\epsilon ={\epsilon }_{m_k}$ converging to a solution $u(x,t)$ to Boundary Value Problem II. Obviously, the limit function $u(x,t)$ belongs to the desired class.

The theorem is proved. □

5. Comments and Supplements

The approach to the study of the solvability of boundary value problems with involution in this article is firstly new and, secondly, does not require that the differential equation has a special form and, finally, it can be used for studying the solvability of boundary value problems for a wide class of differential equations with an involution of a general form and variable coefficients.

Some illustrating examples are as follows.

Observe first of all that, in Boundary Value Problems I and II, the Dirichlet conditions (2) can be replaced by other conditions, for example, by the conditions of the third boundary value problem

$$u_x(0,t)+\alpha \left(t\right)u(0,t)=0,u_x(1,t)+\beta \left(t\right)u(1,t)=0,$$

or by mixed conditions.

Furthermore, Equations (1) and (4) can contain summands with the first derivatives (with respect to x in (1), with respect to x and t in (4)) with variable coefficients and an involution.

It is easy to apply the approach of the present article to equations of an order higher than 2. For example, it is possible to investigate the solvability of boundary value problems for higher-order parabolic equations

$${(-1)}^{m+1}{u}_t(x,t)-\frac{{\partial }^{2m}u(x,t)}{\partial x^{2m}}-a(x,t)\frac{{\partial }^{2m}u(\phi \left(x\right),t)}{\partial x^{2m}}+c(x,t)u(x,t)=f(x,t)$$

and many other equations.

To specify, in fact, we obtained theorems on the solvability of Boundary Value Problems I${}^{\prime }$ and II${}^{\prime }$ for Equations (8) and (9) with an involution at higher derivatives with respect to t.

One more remark: For $a=a\left(t\right)$, degeneration in (1) and (4) is defined by conditions (25) and (28); the existence of regular solutions in the presence of degeneration is guaranteed by additional smoothness of the right-hand side $f(x,t)$. At the same time, the proofs of Theorems 3 and 4 imply that under condition (25) and the condition

$$\sqrt{{\phi }_1}\underset{[0,T]}{max}\left|a\left(t\right)\right|<1$$

(38)

any additional smoothness of $f(x,t)$ is not required. In other words, the presence of degeneration under condition (25) in Equations (1) and (4) does not influence the solvability of the boundary value problems in the presence of condition (38) for the involution $\phi \left(x\right)$.

Note that for the linear-fractional involution given at the beginning of the article, it is not hard to find the reals b and c for which condition (38) will hold for any a priori given function $a\left(t\right)$. Additionally, conversely, for any involution $\phi \left(x\right)$ satisfying (12), the set of the functions $a\left(t\right)$ satisfying (38) is not empty.