UDC 517.944

MATHEMATICS

V. M. BOROK

CLASSES OF WELL-POSED SOLVABILITY OF A BOUNDARY-VALUE PROBLEM IN AN INFINITE LAYER

(Presented by Academician I. G. Petrovskii, 25 IV 1968)

We shall consider equations of the form

\[ \partial^2 u(x,t)/\partial t^2 + P(\partial/\partial x)\partial u(x,t)/\partial t + Q(\partial/\partial x)u(x,t)=0, \tag{1} \]

\[ 0<t<T,\qquad x=(x_1,\ldots,x_n),\qquad -\infty<x_i<\infty,\qquad i=1,\ldots,n, \]

with boundary conditions

\[ u(x,0)=u_0(x),\qquad u(x,T)=u_T(x),\qquad -\infty<x_i<\infty,\qquad i=1,\ldots,n. \tag{2} \]

Here \(P(\partial/\partial x), Q(\partial/\partial x)\) are polynomials in
\(\partial/\partial x_1,\ldots,\partial/\partial x_n\) with constant coefficients; \(u_0(x), u_T(x)\) are prescribed functions. By a solution of problem (1)—(2) we shall mean (continuous) functions \(u(x,t)\) satisfying equation (1) and conditions (2) in the sense of generalized functions \(\left({}^{1}, \text{pp. }129\text{—}130\right)\). We investigate the question: under what conditions on the smoothness, as well as on the growth as \(|x|\to\infty\), of the functions \(u_0(x)\) and \(u_T(x)\) and their derivatives does a solution of problem (1)—(2) (in the above sense) exist and is unique* in one or another class of functions, and also depend continuously on the functions \(u_0(x)\) and \(u_T(x)\) in a sense that will be specified each time.

\(1^\circ\). Definition 1. Problem (1)—(2) is called well-posedly solvable in the class of bounded functions if:
a) for any \(l\ge 0\) there exists an \(l_1\ge 0\) such that, for any functions \(u_0(x)\) and \(u_T(x)\) bounded for all \(x\in R_n\) together with all their derivatives up to order \(l_1\), there exists a unique solution \(u(x,t)\) of problem (1)—(2), which for each \(t\in(0,T)\) is bounded together with all its derivatives up to order \(l\);
b) if the functions \(u_{0m}(x)\) and \(u_{Tm}(x)\) and their derivatives up to order \(l_1\) tend (uniformly in \(R_n\)) to zero as \(m\to\infty\), then the corresponding solutions \(u_m(x,t)\) of problem (1)—(2), for each \(t\in(0,T)\), also tend uniformly to zero together with their derivatives up to order \(l\).

Introduce the following notation. Let \(Z\) be the set of those points \(s=\sigma+i\tau=(\sigma_1+i\tau_1,\ldots,\sigma_n+i\tau_n)\) for which

\[ D(s)\equiv \frac14 P^2(-is)-Q(-is)=-k^2\pi^2/T^2 \]

for some integer \(k\ge 1\). Denote

\[ \Lambda(s)=|\operatorname{Re} P(is)|-2|\operatorname{Re}\sqrt{D(-s)}|. \]

Definition 2. Equation (1) will be called correct if:
a) real values \(s=\sigma\) do not belong to the set \(Z\);
b) the function \(\Lambda(s)\) is bounded above for all real \(s=\sigma\): \(\Lambda(\sigma)\le C\), \(-\infty<\sigma_i<\infty\), \(i=1,\ldots,n\). For example, any equation (1) for which \(P(s)\equiv 0\) (or \(P(s)\equiv \mathrm{const}\)) will be correct if condition a) is fulfilled for it. Thus the equations

\[ \partial^2 u/\partial t^2-\partial^2 u/\partial x^2+\partial u/\partial x=0,\qquad \partial^2 u/\partial t^2+\partial^2 u/\partial x^2=0, \]

\[ \partial^2 u/\partial t^2\pm \partial u/\partial x=0 \tag{1′} \]

* The question of uniqueness of the solution of problem (1)—(2) is the subject of the author’s note \(\left({}^{2}\right)\).

are correct, while the equation \(\partial^2 u/\partial t^2-\partial^2 u/\partial x^2=0\) is not correct.

The naturalness of singling out the class of correct equations is shown by the following

Theorem 1. In order that problem (1)—(2) be correctly solvable in the class of bounded functions, it is necessary and sufficient that equation (1) be correct.

\(2^\circ\). One can also indicate broader classes of functions (in the sense of growth at infinity) in which problem (1)—(2) for a correct equation (1) is correctly solvable.

Lemma 1. If equation (1) is correct, then there exist constants \(A>0\) and \(\alpha\) such that in the domain \(\Delta_{\alpha,A}\)

\[ \Delta_{\alpha,A}=\{s=\sigma+i\tau:\ |\tau|\le A(1+|\sigma|)^\alpha,\ -\infty<\sigma_i<\infty,\ i=1,\ldots,n\} \tag{3} \]

two conditions are satisfied: 1) \(\Delta_{\alpha,A}\cap Z=\varnothing\); 2) there exists \(C_1\) such that \(\Lambda(s)\le C_1\) for \(s\in\Delta_{\alpha,A}\).

Definition 3. We shall call a correct equation (1) an equation of zero type if the result of Lemma 3 holds with \(\alpha\ge 0\), and an equation of negative type otherwise. We shall say that a correct equation of negative type has type \(\alpha_0\) \((<0)\), if \(\alpha_0\) is the exact upper bound of those values of \(\alpha\) for which the result of Lemma 1 is valid in \(\Delta_{\alpha,A}\).

Denote by \(M_{\beta,B}\) the class of (locally integrable) functions \(f(x)\) satisfying the estimate

\[ |f(x)|\le C_f\exp\{B|x|^\beta\}. \tag{4} \]

Theorem 2. Let (1) be a correct equation of zero type, and let \(A_0>0\) be the exact upper bound of those \(A\) for which, in the domain \(\Delta_{0,A}\), the result of Lemma 1 is valid. Then for any \(l\ge 0\) one can indicate an \(l_1\ge 0\) such that, for any functions \(u_0(x)\), \(u_T(x)\) which belong to the class \(M_{1,B}\) \((B<A_0)\) together with all their derivatives up to order \(l_1\), there exists in the class \(M_{1,B}\) a unique solution \(u(x,t)\) of problem (1)—(2); this solution has derivatives up to order \(l\) (with respect to \(x\)) belonging to \(M_{1,B}\) and depends continuously on the functions \(u_0(x)\), \(u_T(x)\) in the following sense: if in the estimate

\[ \max_{k\le l_1}\max\left\{\left|\partial^k u_{0m}(x)/\partial x_1^{k_1}\cdots\partial x_n^{k_n}\right|,\ \left|\partial^k u_{Tm}(x)/\partial x_1^{k_1}\cdots\partial x_n^{k_n}\right|\right\} \le C_m\exp\{B|x|\}, \]

\[ m=1,2,\ldots, \]

\(C_m\to0\) as \(m\to\infty\), then in the corresponding estimate of the solution and its derivatives

\[ \max_{k\le l}|D^{(k)}u_m(x,t)|\le C_{mt}\exp\{B|x|\},\quad t\in(0,T), \]

also \(C_{mt}\to0\) as \(m\to\infty\).

For example, the first of equations \((1')\) is a correct equation of zero type. For it \(A_0=1/2\), if \(T\ge 2\pi\), and \(A_0=1/2-\sqrt{1/4-\pi^2/T^2}\), if \(0<T\le 2\pi\).

Theorem 3. Let (1) be a correct equation of negative type. Then for any \(l\ge 0\) and \(m\ge 0\) there exists an \(l_1\ge 0\) such that if the functions \(u_0(x)\), \(u_T(x)\) have polynomial order of growth \(m\) together with all their derivatives up to order \(l_1\):

\[ \max_{k\le l}\max\{|D^{(k)}u_0(x)|,\ |D^{(k)}u_T(x)|\}\le C(1+|x|)^m, \tag{5} \]

then problem (1)—(2) has a (unique) solution \(u(x,t)\), which together with its derivatives up to order \(l\) has polynomial order of growth not exceeding \(m+n+1\):

\[ \max_{k\le l}|D^{(k)}u(x,t)|\le C_t(1+|x|)^{m+n+1},\quad 0<t<T. \tag{6} \]

This solution depends continuously on the initial functions \(u_0(x)\), \(u_T(x)\) in the sense that \(C_t\) in (6) depends on \(C\) in (5) and tends to zero as \(C \to 0\).

\(3^\circ\). Definition 4. We shall call the correct equation (1) strongly correct if there exist constants \(C_1>0\), \(h>0\), and \(C_2\) such that

\[ \Lambda(\sigma)\leq -C_1|\sigma|^h+C_2,\qquad -\infty<\sigma_i<\infty,\quad i=1,\ldots,n. \tag{7} \]

For example, the last two of equations \((1')\) are strongly correct; for the first of them \(h=1\), for the second \(h=\tfrac12\).

For strongly correct equations there is a result analogous to that formulated in Lemma 1.

Lemma 2. There exists a domain \(\Delta_{\alpha,A}\) of the form (3) such that \(\Delta_{\alpha,A}\cap Z=\varnothing\), and in \(\Delta_{\alpha,A}\) the estimate (7) is preserved, possibly with \(C_1\) and \(C_2\) replaced by some other constants \(\hat C_1>0\) and \(\hat C_2>0\), and \(h\) by any \(h_1<h\).

This fact makes it possible, analogously to Definition 3, to introduce a classification of strongly correct equations, distinguishing equations of zero kind and equations of negative kind \(\alpha_0\).

Definition 5. We shall say that the problem (1)—(2) is correctly solvable in the class of functions \(M_{\beta,B}\) if: 1) for any functions \(u_0(x)\in M_{\beta,B}\) and \(u_T(x)\in M_{\beta,B}\) there exists a unique solution \(u(x,t)\) of problem (1)—(2), which for each \(t\in(0,T)\) is a function of the class \(M_{\beta,B}\); 2) if
\[ |u_{0m}(x)|\leq C_m\exp\{B|x|^\beta\},\qquad |u_{Tm}(x)|\leq C_m\exp\{B|x|^\beta\} \]
and \(C_m\to0\) as \(m\to\infty\), then the corresponding solutions \(u_m(x,t)\) of problem (1)—(2) satisfy the estimates
\[ |u_m(x,t)|\leq C_{tm}\exp\{B|x|^\beta\}, \]
where \(C_{tm}\to0\) as \(m\to\infty\) \((0<t<T)\).

Theorem 4. Let (1) be a strongly correct equation of zero kind, and let \(A_0>0\) be the exact upper bound of those \(A\) for which, in the domain \(\Delta_{0,A}\), the result of Lemma 2 is valid. Then the problem (1)—(2) is correctly solvable in any class \(M_{1,B}\), where \(B<A_0\), and also in any class \(M_{\beta,B}\), where \(0<\beta<1\), \(B>0\) is arbitrary.

For example, the second and third equations in (1) belong to the class of strongly correct equations of zero kind. For the first of them \(A_0=\pi/T\), for the second \(A_0=\pi^2/T^2\).

Theorem 5. Let equation (1) be a strongly correct equation of negative kind \(\alpha\). Then the problem (1)—(2) is correctly solvable in the class of functions \(M_{\beta,B}\) for \(\beta<h/(h-\alpha)\) and any \(B>0\).

\(4^\circ\). Theorems 4 and 5 show that correct solvability of the problem (1)—(2) for strongly correct equations takes place in classes of increasing functions without assumptions concerning the smoothness of the functions \(u_0(x)\) and \(u_T(x)\). It can be shown that in this case the solution of the problem (1)—(2) turns out to be an (infinitely) differentiable (with respect to \(x\)) function for \(t\in(0,T)\). This property proves to be characteristic for strongly correct equations. Namely, the following holds.

Theorem 6. If, with respect to an equation of the form (1) (for \(n=1\)), it is known that: a) the problem (1)—(2) for it is correctly solvable in some class of functions \(M\), containing all bounded functions of the first Baire class, i.e., for any \(u_0(x)\in M\) and \(u_T(x)\in M\) there exists one and only one solution \(u(x,t)\in M\), and from the fact that the sequences
\[ u_{0m}(x)\longrightarrow 0,\qquad u_{Tm}(x)\longrightarrow 0 \quad (m\to\infty) \]
uniformly on any interval and bounded on the whole line, it follows that the corresponding sequence of solutions
\[ u_m(x,t)\longrightarrow 0 \]
in the same sense; b) the solution \(u(x,t)\) of the problem (1)—(2) for any \(t>0\) is a continuously differentiable function, then equation (1) is strongly correct.

\(5^\circ\). We now single out those correct equations for which correct solvability of the problem (1)—(2) takes place in the class of all sufficiently smooth functions without restriction on their growth as \(|x|\to\infty\).

Theorem 7. In order that problem (1)—(2) have a unique solution \(u(x,t)\) for arbitrary functions \(u_0(x)\) and \(u_T(x)\) possessing derivatives up to some order \(l\), and that from the fact that the sequences of functions \(u_{0m}(x)\), \(u_{Tm}(x)\) tend to zero as \(m \to \infty\), together with all their derivatives up to order \(l\), uniformly in every bounded domain, it should follow that the corresponding sequence of solutions \(u_m(x,t)\), for each \(t \in (0,T)\), also tends uniformly to zero in every bounded domain as \(m \to \infty\), it is necessary and sufficient that equation (1) have the form

\[ \left(\frac{\partial}{\partial t}+\sum_{k=1}^{n} A_k \frac{\partial}{\partial x_k}+B\right) \left(\frac{\partial}{\partial t}+\sum_{k=1}^{n} A_k \frac{\partial}{\partial x_k}+C\right)u(x,t)=0, \]

where \(A_1,\ldots,A_n\) are arbitrary real constants, and \(B\) and \(C\) are complex constants, \(T(B-C)\ne 2k\pi i,\ k=\pm1,\pm2,\ldots\).

Let us finally note that if equation (1) is not well-posed, then the existence of a solution of problem (1)—(2) can be guaranteed under the assumption that the functions \(u_0(x)\) and \(u_T(x)\) are continued into the complex space \(z=x+iy\) in such a way that entire functions of exponential type are obtained, depending on equation (1).

Kharkov State University
named after A. M. Gorky

Received
24 IV 1968

REFERENCES

1. I. M. Gel'fand, G. E. Shilov, Generalized Functions, vol. 3, Moscow, 1958.
2. V. M. Borok, DAN, 183, No. 5 (1968).