Abstract Generated abstract
The paper studies boundary-value problems for elliptic-parabolic differential equations in bounded cylinders, with an elliptic spatial operator and a finite-order time differential operator. Using localization, eigenfunction expansions, Sobolev-type spaces, and parameter-dependent elliptic estimates, it establishes isomorphism and a priori estimates in the self-adjoint case, then proves Fredholm solvability for semibounded operators under Shapiro-Lopatinskii-type conditions. The results include existence and uniqueness criteria for problems with initial and terminal boundary conditions in time, as well as a uniqueness theorem for an analogue of the Dirichlet problem and illustrative higher-order examples.
Full Text
Reports of the Academy of Sciences of the USSR
1969. Volume 184, No. 3
UDC 517.946
MATHEMATICS
Yu. A. DUBINSKII
ON BOUNDARY-VALUE PROBLEMS FOR ELLIPTIC-PARABOLIC EQUATIONS
(Presented by Academician I. G. Petrovskii on 29 III 1968)
In this paper* boundary-value problems are studied for equations of the form
\[ \mathfrak{A}(u)\equiv P_s(t,x,\partial/\partial t)u+L(t,x,D)u=h(t,x), \tag{1} \]
where
\[ P_s\left(t,x,\frac{\partial}{\partial t}\right)u = \sum_{q=0}^{s} a_q(t,x)u^{(q)},\qquad u^{(q)}=\frac{\partial^q u}{\partial t^q},\qquad s\geqslant 1; \]
\[ L(t,x,D)u\equiv \sum_{|\alpha|\leqslant 2m} a_\alpha(t,x)D^\alpha u \]
is an elliptic differential operator of order \(2m\). The coefficients \(a_q(t,x)\) and \(a_\alpha(t,x)\) are smooth complex functions, \(a_s=\pm 1\). In papers \((^{2,3})\) we considered boundary-value problems for equations of the form (1) in a half-infinite cylinder, and also the problem of periodic solutions with respect to \(t\). Here boundary-value problems for equation (1) are studied in the bounded cylinder \(Q=G\times[0,T]\), where \(G\subset \mathbf{R}^n\) is a domain with smooth boundary.
Definition. The operator \(\mathfrak{A}(u)\) is called elliptic-parabolic (e.-p.) if, for all real \(\tau\in \mathbf{R}^1\) and \(\xi\in \mathbf{R}^n\) such that \(|\tau|+|\xi|\ne 0\),
\[ a_s(i\tau)^s+L_{2m}(t,x,i\xi)\ne 0, \]
where \(L_{2m}(t,x,i\xi)\) is the principal part of the polynomial \(L(t,x,i\xi)\).
For e.-p. equations the localization principle is valid. Therefore these equations are first studied in the unbounded cylinder \(Q=G\times(-\infty,+\infty)\), then in the cylinder \(Q=G\times[0,\infty)\), and finally, by means of a partition of unity, in \(Q=G\times[0,T]\).
§ 1. The case of a self-adjoint operator. In this paragraph \(L(t,x,D)\equiv L(x,D)\). Suppose there is a system of boundary conditions
\[ B_j(x,D)u|_{\partial G}=0,\qquad j=1,\ldots,m, \tag{2} \]
where
\[ B_j(x,D)u\equiv \sum_{|\beta|\leqslant m_j} b_{\beta j}(x)D^\beta u \]
are differential operators of orders \(m_j\leqslant 2m-1\), defining together with the operator \(L(x,D)u\) a self-adjoint boundary-value problem. Denote by \(H^l\{B_j\}\) the space of Fourier series in the eigenfunctions \(\omega_0(x),\omega_1(x),\ldots\) of this problem, converging in the metric of \(H^l\) (see \((^{4,5})\)).
Next, denote by \(H(r,l)\) the space of functions \(u(t,x)\) defined in the cylinder \(Q=G\times(a,b)\), with norm
\[ \|u\|_{r,l}^2=\int_a^b\left(\|u^{(r)}\|_0^2+\|u\|_l^2\right)\,dt. \tag{3} \]
The space of Fourier series
\[ u(t,x)=c_0(t)\omega_0(x)+c_1(t)\omega_1(x)+\cdots, \]
converging in \(H(r,l)\), will be denoted by \(H(r,l)\{B_j\}\).
* A detailed exposition is being published in the journal “Izv. AN Armenian SSR, Ser. Mathematics.”
Theorem 1. Let \(Q=G\times \mathbf{R}^{1}\), \(a_q=0\), \(0\leq q\leq s-1\). If the operator \(\mathfrak{A}(u)\) is e.-p. and the first eigenvalue \(\lambda_0\) of the operator \(L(x,D)u\) is positive, i.e. \(\lambda_0>0\), then it maps the space \(H(r,r2m/s)\{B_j\}\) isomorphically onto the space \(H(r-s,(r-s)2m/s)\{B_j\}\). Moreover,
\[ \|h\|_{r-s,(r-s)2m/s}\leq C_1\left(\|u\|_{r,r2m/s}+\lambda_0^{r/s}\|u\|_{0,0}\right)\leq C_2\|h\|_{r-s,(r-s)2m/s}. \]
We now consider equation (1) in the semi-infinite cylinder \(Q=G\times[0,\infty)\). It is required to find a solution of equation (1) under the initial conditions
\[ u^{(n_i)}(0,x)=\psi_i(x),\qquad i=0,\ldots,s-1-k, \tag{4} \]
where \(k=[s/2]\), if \(s\) is odd and \((-1)^{[s/2]}a_s>0\); \(k=[s/2]+1\), if \(s\) is odd and \((-1)^{[s/2]}a_s<0\); \(k=s/2\), if \(s\) is even.
Assume that for every \(\lambda>0\) the determinant
\[ \det\|\mu_j^{\,n_i}(\lambda)\|\neq 0,\qquad i,j=0,\ldots,s-1-k, \tag{5} \]
where \(\mu_j(\lambda)\) are the roots of the algebraic equation \(a_s\mu^s+\lambda=0\) such that \(\operatorname{Re}\mu_j(\lambda)<0\).
Theorem 2. Let \(Q=G\times[0,\infty)\), \(a_q=0\), \(0\leq q\leq s-1\); let the operator \(\mathfrak{A}(u)\) be e.-p. and let condition (5) be satisfied. Then, if the first eigenvalue \(\lambda_0\) of the operator \(L(x,D)u\) is positive, i.e. \(\lambda_0>0\), then for any function \(h(t,x)\in H(r-s,(r-s)2m/s)\{B_j\}\) and any functions \(\psi_i(x)\in H^{l_i}\{B_j\}\), \(l_i=(r-n_i)2m/s-m/s\), there exists a unique solution \(u(t,x)\in H(r,r2m/s)\{B_j\}\) satisfying conditions (2), (4). Moreover,
\[ A(h,\psi)\leq C_1\left(\|u\|_{r,r2m/s}+\lambda_0^{r/s}\|u\|_{0,0}\right)\leq C_2 A(h,\psi), \]
where
\[ A(h,\psi)\equiv \|h\|_{r-s,(r-s)2m/s} +\sum_{i=0}^{s-1-k}\|\psi_i\|_{l_i},\qquad r\geq \max(2m,n_i+1). \]
We now consider equation (1) with variable coefficients \(a_q(t,x)\) in the bounded cylinder \(Q=G\times[0,T]\). It is required to find a solution under the additional conditions
\[ \sum_{q=0}^{n_i} p_{iq}(t,x)u^{(q)}\big|_{t=0}=\psi_i(x),\qquad i=0,\ldots,s-1-k,\qquad p_{in_i}\equiv 1; \tag{6} \]
\[ \sum_{q=0}^{n_l} r_{lq}(t,x)u^{(q)}\big|_{t=T}=\chi_l(x),\qquad l=0,\ldots,k-1,\qquad r_{ln_l}\equiv 1. \tag{7} \]
Let, for sufficiently large \(\lambda>0\), the following condition be satisfied.
Condition 1 (the Shapiro–Lopatinskii condition):
1) \(\det\|\mu_j^{\,n_i}(\lambda)\|\neq 0\), \(i,j=0,\ldots,s-1-k\), where \(\mu_j(\lambda)\) are the roots of the equation \(a_s\mu^s+\lambda=0\) such that \(\operatorname{Re}\mu_j(\lambda)<0\);
2) \(\det\|\mu_j^{\,n_l}(\lambda)\|\neq 0\), \(l,j=0,\ldots,k-1\), where \(\mu_j(\lambda)\) are the roots of the equation \(a_s\mu^s+\lambda=0\) such that \(\operatorname{Re}\mu_j(\lambda)>0\).
Theorem 3. Let condition 1 be satisfied and let \(\lambda_0>0\) be sufficiently large. Then problem (1), (2), (6), (7) is uniquely solvable, i.e. for any functions \(h(t,x)\in H(r-s,(r-s)2m/s)\{B_j\}\), \(\psi_i(x)\in H^{(r-n_i)2m/s-m/s}\{B_j\}\), \(\chi_l(x)\in H^{(r-n_l)2m/s-m/s}\{B_j\}\), there exists a unique solution of this problem \(u(t,x)\in H(r,r2m/s)\{B_j\}\). Moreover, the estimate
\[ A(h,\psi,\chi)\leq C_1\left(\|u\|_{r,r2m/s}+\lambda_0^{r/s}\|u\|_{0,0}\right)\leq C_2 A(h,\psi,\chi), \]
where
\[ A(h,\psi,\chi)\equiv \|h\|_{r-s,(r-s)2m/s} +\sum_{i=0}^{s-1-k}\|\psi_i\|_{(r-n_i)2m/s-m/s} +\sum_{l=0}^{k-1}\|\chi_l\|_{(r-n_l)2m/s-m/s}. \]
Corollary. If condition 1 is satisfied, then problem (1), (2), (6), (7) is Fredholm, i.e., the kernel and cokernel of the problem are finite-dimensional, and their dimensions are equal.
§ 2. The case of the semibounded operator \(L(t,x,D)u\)
Suppose that the following conditions are satisfied:
I. For every fixed \(t\in[0,T]\) the operator \(L(t,x,D)u\) is bounded from below, i.e.,
\[ \lambda_0(t)=\inf_{u\in H^{2m}\{B_j\}} \frac{\operatorname{Re}\langle L(t,x,D)u,\bar u\rangle}{\langle u,\bar u\rangle}>-\infty . \]
II. For any fixed \(t\in[0,\varepsilon]\) \((t\in[T-\varepsilon,T])\), where \(\varepsilon>0\) is some number, the operator \(L(t,x,D)u\equiv L(0,x,D)u\) \((L(t,x,D)u\equiv L(T,x,D)u)\) is a self-adjoint operator in the space \(H^{2m}\{B_j\}\).
III. Condition 1 of § 1 is satisfied.
IV. The coefficients of the operator \(\mathfrak A(u)\) and of the boundary operators (6), (7) are sufficiently smooth, and moreover \(a_q(t,x)\equiv a_q(t)\), \(t\in[0,\varepsilon]\vee[T-\varepsilon,T]\).
Before formulating the main theorem of this paragraph, we introduce the necessary spaces. Denote by \(\hat H(r,r2m/s)\) the space of functions \(u(t,x)\in H(r,r2m/s)\) satisfying conditions (2) and such that, if \(\varphi(t)\in C^\infty[0,T]\) and \(\operatorname{supp}\varphi(t)\cap[0,\varepsilon]\ne\varnothing\) or \(\operatorname{supp}\varphi(t)\cap[T-\varepsilon,T]\ne\varnothing\), then \(\varphi(t)u(t,x)\in H(r,r2m/s)\{B_j\}\). In other words, it is required of functions \(u(t,x)\in \hat H(r,r2m/s)\) that in some neighborhoods of the points \(t=0\) and \(t=T\) they be expandable in the eigenfunctions of the operators \(L(0,x,D)\) and \(L(T,x,D)\), respectively. Otherwise they are arbitrary.
Theorem 4. If conditions I—IV are satisfied, then problem (1), (2), (6), (7) is Fredholm. This means that for any functions \(h(t,x)\in \hat H(r-s,(r-s)2m/s)\), \(\psi_i(x)\in H^{(r-n_i)2m/s-m/s}\{B_j\}\), \(i=0,\ldots,s-1-k\), \(\chi_l(x)\in H^{(r-n_l)2m/s-m/s}\{B_j\}\), \(l=0,\ldots,k-1\), subject to a finite number \(\varkappa\) of conditions, there exists a solution of this problem \(u(t,x)\in \hat H(r,r2m/s)\). At the same time it is determined up to the addition of any solution of the homogeneous problem, among which there are exactly \(\varkappa\) linearly independent ones. If \(\lambda_0=\min\lambda_0(t)\), \(0\le t\le T\), is sufficiently large, then \(\varkappa=0\), i.e., there is an isomorphism. Moreover, the corresponding estimates hold.
In proving this theorem, estimates for elliptic problems with a parameter, obtained in [1], are used.
§ 3. Analogue of the Dirichlet problem. Uniqueness theorem. Examples
In this paragraph we consider the simplest example of problem (1), (2), (6), (7). Namely, let
\[ \mathfrak A(u)\equiv P_s\left(\frac{\partial}{\partial t}\right)u+L(x,D)u=h(t,x), \tag{8} \]
where
\[ P_s\left(\frac{\partial}{\partial t}\right)u\equiv \sum_{q=0}^{s}a_qu^{(q)},\qquad a_q\in C^1,\qquad a_s=\pm1, \]
\(L(x,D)u\) is a semibounded operator of order \(2m\).
We seek a solution of equation (8) under conditions (2), (9),
\[ u(0,x)=\cdots=u^{(s-1-k)}(0,x)=u(T,x)=\cdots=u^{(k-1)}(T,x)=0. \tag{9} \]
As follows from Theorem 4, problem (8), (2), (9) is Fredholm. We shall now be interested in the question of the unique solvability of this problem.
Theorem 5 (uniqueness). If the polynomial in \(\tau \in \mathbb{R}^1\)
\[ \operatorname{Re} \sum_{q=0}^{s} a_q (i\tau)^q + \lambda_0 > 0, \]
then problem (8), (2), (9) has a unique solution.
Example 1. Consider the equation
\[ a_3 u''' + (-1)^m \Delta^m u = h(t,x). \]
It is required to find a solution of this equation under conditions (2) such that
\[ u(0,x) = u'(0,x) = u(T,x) = 0, \qquad \text{if } a_3=-1. \]
\[ u(0,x) = u(T,x) = u'(T,x) = 0, \qquad \text{if } a_3=1. \]
The problem has a unique solution.
Example 2. Consider the equation of even order
\[ u^{(2k)} + (-1)^{m-k}\Delta^m u = h(t,x). \]
For this equation, the problem under investigation is the Dirichlet problem (in \(t\)):
\[ u(0,x)=\cdots=u^{(k-1)}(0,x)=u(T,x)=\cdots=u^{(k-1)}(T,x)=0. \tag{10} \]
By Theorems 4 and 5, the Dirichlet problem (10) is well posed. For an analogue of the Dirichlet problem, see \((^6,^7)\).
Moscow Power Engineering Institute
Received
27 III 1968
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