Full Text
UDC 517.946
MATHEMATICS
A. Kh. GUDIEV
ON A BOUNDARY VALUE PROBLEM FOR LINEAR PARABOLIC EQUATIONS WITH DISCONTINUOUS COEFFICIENTS
(Presented by Academician S. L. Sobolev on 18 IX 1967)
In the present article the first boundary value problem is considered for a linear equation of parabolic type
\[ Lu \equiv \frac{\partial u}{\partial t} -\frac{\partial}{\partial x_i}\left(a_{ij}u_{x_j}+a_i u+f_i\right) +b_i u_{x_i}+au+f=0 \tag{1} \]
with unbounded lower-order coefficients and free terms in the cylinder \(Q_T=D\times[0\le t\le T]\) \((T<\infty)\), whose base is a bounded domain \(D\) of the Euclidean space \(E^n\).
We take the initial and boundary conditions in the form
\[ u|_{t=0}=\psi_0(x), \qquad u|_{S_T}=0. \tag{2} \]
Numerous works have been devoted to the study of existence and uniqueness of the generalized solution of problem (1), (2) in one class or another (see the detailed bibliography in \((^{1-5})\)). In the monograph \((^1)\) the latest results in this direction are set forth in detail. In the case when \(a_i^2\), \(b_i^2\), \(a\) with respect to the variable \(x\) belong to \(L_p\), \(p>n\), in \((^1)\) conditions are given ensuring uniqueness and existence of a generalized solution from \(V_2\) of problem (1), (2); here the usual embedding theorems are used essentially.
However, in the case \(p<n\) these conditions are insufficient for the existence and uniqueness of a generalized solution from \(V_2\) of problem (1), (2). In this case, some additional conditions are needed for unique solvability. Moreover, in the case \(p<n\) the usual theorems do not work.
In the present article, in the case \(1<p<n\), in terms of spaces with mixed norm, conditions on the data of the problem are found under which the unique solvability of problem (1), (2) in \(V_2\) holds. In addition, the smoothness of the solution \(u(x,t)\) with respect to \(t\) is studied (for the definitions of the spaces \(V_2\), \(V_2^{1,0}\), \(V_2^{*}\), see \((^1)\)).
Let \(s\) be a fixed natural number not exceeding \(n\); \(E^n\) is the \(n\)-dimensional Euclidean space of points \(\bar x\); \(E^{n+1}\) is the \((n+1)\)-dimensional Euclidean space of points \((\bar x,t)\); \(E^s(E^{n-s})\) is the \(s\)-dimensional \(((n-s)\)-dimensional) space of points \(\bar x_s(\bar x_{n-s})\), where \(\bar x=(\bar x_s,\bar x_{n-s})\), \(\bar x_s=(x_1,x_2,\ldots,x_s)\), \(\bar x_{n-s}=(x_{s+1},x_{s+1},\ldots,x_n)\); \(D\) is a bounded domain of class \(C(\bar h,\omega)\) in \(E^n\) (for the definition of the class \(C(\bar h,\omega)\), see \((^2)\)); \(Q_T\) is the cylinder \(D\times[0\le t\le T]\); \(D_1=D\cap(\bar x_{n-s}=\mathrm{const})\); \(D_2=\operatorname{pr}_{E^{\,n-s}}D\); \(L_{(p_1,p_2,l)}(Q_T)\) is the Banach space consisting of all functions measurable on \(Q_T\) and having finite norm
\[ \|f\|_{L_{(p_1,p_2,l)}(Q_T)} =\bigl\|\|f\|_{L_{(p_1,p_2)}(D)}\bigr\|_{L_l(0\le t\le T)}. \]
Let \(\Omega^{(\varepsilon)}\) denote the following set of points \((r_1,r_2)\) of the plane \(r_1Or_2\):
\[ \Omega_k^{(\varepsilon)} \equiv \left\{ \begin{array}{l} (r_1,r_2):\quad kr_1r_2-(n-s)r_1-sr_2-\varepsilon=0;\\[2mm] \infty>r_1> \begin{cases} 1, & \text{if } s<k,\\ s/k, & \text{if } s\geq k; \end{cases}\\[3mm] \infty<r_2> \begin{cases} 1, & \text{if } n-s<k,\\ (n-s)/k, & \text{if } n-s\geq k; \end{cases}\\[3mm] r_1\geq r_2. \end{array} \right\}. \]
Put, for any real positive number \(p\geq 1\),
\[ \Omega_{k;p}^{(\varepsilon)}\equiv \Omega_k^{(\varepsilon)}\cap \{r_1,r_2\geq p\}. \]
Obviously, \(\Omega_{k;1}^{(\varepsilon)}=\Omega_k^{(\varepsilon)}\). Put \(\Omega_{k;p}^{(0)}=\Omega_{k;p}\). The class \(X_{(m-2/l_1)}\) of Banach spaces \(L_{(r_1,r_2,l)}(Q_T)\) is defined by the equality
\[ X_{(m-2/l_1)}\equiv \{L_{(r_1,r_2,l_1)}(Q_T);\ (r_1,r_2)\in \Omega_{(m-2/l_1)},\ l_1>1\}. \]
We define the class \(\widehat X_{(2-2/l_1);p}\) by the equality
\[ \widehat X_{(2-2/l_1);p} = \bigcup_{(r_1,r_2)\in \Omega_{(2-2/l_1);p}} L_{(r_1,r_2,l_1)}(Q_T). \]
Let
\[ R_{ij}=R_{ij}(r_{ij1},r_{ij2},l_1),\qquad P_{ij}=P_{ij}(p_{ij1},p_{ij2},l),\quad i=1,2,\ldots,n;\quad j=1,2, \]
where \(p_{ijk}=2r_{ijk}\), \(k=1,2\); \(l=2l_1\),
\[ R'_{ij}=R'_{ij}(r'_{ij1},r'_{ij2},l'_1),\qquad P'_{ij}=P'_{ij}(p'_{ij1},p'_{ij2},l'), \]
\[ R_3=R_3(r_{31},r_{32},l_1),\qquad R_4=R_4(r_{41},r_{42},l'), \]
\[ P_3=P_3(p_{31},p_{32},l),\qquad P_4=P_4(p_{41},p_{42},l),\qquad p_{kj}=2r_{kj}. \]
Theorem 1. If \(u(x,t)\in \dot V_2(Q_T)\) and the numbers \(q_1,q_2,l\) satisfy the conditions
\[ 1/l+s/2q_1+(n-s)/2q_2=n/4,\qquad 2\leq q_1,q_2,l, \]
then
\[ \|u\|_{L(q_1,q_2,l)(Q_T)} \leq c\, \operatorname*{vrai\,max}_{0\leq t\leq T} \|u\|_{L_2(D)}^{\,1-2/l}\, \|u_x\|_{L_2(Q_T)}^{\,2/l}. \]
Corollary. Under the condition of Theorem 1, the estimate
\[ \|u\|_{L(q_1,q_2,l)(Q_T)}\leq c\,|u|_{Q_T} \]
holds.
Lemma 1. If the numbers \(q_1,q_2,l\) satisfy the conditions
\[ 1/l+s/2q_1+(n-s)/2q_2>n/4, \tag{3} \]
then one can choose numbers \(\widetilde q_1,\widetilde q_2,\widetilde l\) satisfying the conditions
\[ 1/\widetilde l+s/2\widetilde q_1+(n-s)/2\widetilde q_2=n/4,\qquad \widetilde l/\widetilde q_2=l/q_2, \]
\[ 1/q_1-1/q_2=1/\widetilde q_1-1/\widetilde q_2. \tag{4} \]
Lemma 2. If the numbers \(q_1,q_2,l\) and \(\widetilde q_1,\widetilde q_2,\widetilde l\) satisfy conditions (3), (4) and \(f\in L_{(\widetilde q_1,\widetilde q_2,\widetilde l)}(Q_T)\), then the estimate
\[ \|f\|_{L(q_1,q_2,l)(Q_T)} \leq \|f\|_{L(\widetilde q_1,\widetilde q_2,\widetilde l)(Q_T)} \left\{ \int_0^T [\operatorname{mes} A(t)]^{\widetilde l/q}\,dt \right\}^{(\widetilde l-l)/\widetilde l l}; \]
\(A(t)\) is the set of points \(x\in D\) where \(|f(x,t)|>0\).
Consider equation (1). Suppose the coefficients satisfy the conditions
\[ \nu \xi_i \xi_i \leq a_{ij}\xi_i\xi_j \leq \mu \xi_i\xi_i,\qquad \nu,\mu=\operatorname{const}>0,\qquad a_{ij}=a_{ji}; \tag{5} \]
\[ \|a_i^2\|_{L_{R_{i1}}(Q_T)},\quad \|b_i^2\|_{L_{R_{i2}}(Q_T)},\quad \|a\|_{L_{R_3}(Q_T)}\leq \mu_1; \tag{6} \]
\[ \left\|\left(\sum f_i^2\right)^{1/2}\right\|_{L_2(Q_T)},\quad \|f\|_{L_{R_4}(Q_T)}\leq \mu_1, \tag{7} \]
where
\[ L_{R_3}(Q_T),\; L_{R_{ij}}(Q_T)\in X_{(2-2/l_1)},\qquad i=1,2,\ldots,n;\quad j=1,2; \tag{8} \]
\[ L_{R_4}\in X_{((n+4)/2-2/l')}. \tag{9} \]
Denote
\[ L_1(u,\eta)\equiv \int_D \bigl[(a_{ij}u_{x_j}+a_i u)\eta_{x_i}+(b_i u_{x_i}+au)\eta\bigr]\,dx, \]
\[ L_2(\hat f,\eta)\equiv \int_D (f\eta+f_i\eta_{x_i})\,dx. \]
Theorem 2. Suppose \(u(x,t)\in \mathring V_2(Q_T)\), and \(u\) satisfies, for almost all \(t_1\) and \(t_2\) in \([0,T]\), including for \(t_1=0\), the inequalities
\[ -\frac12\int_D u^2(x,t)\,dx\Big|_{t=t_1}^{t=t_2} +\int_{t_1}^{t_2}\bigl[L_1(u,u)+L_2(\hat f,u)\bigr]\,dt\leq 0. \tag{10} \]
The coefficients \(a_{ij}, b_i, a_i, a\) and the free terms \(f\) and \(f_i\) satisfy conditions (5)—(9). Then
\[ |u|_{Q_T}\leq C\bigl[\|u(x,0)\|_{L_2(D)} +\|\bar f\|_{L_2(Q_T)}+\|f\|_{L_{R_4}(Q_T)}\bigr], \tag{11} \]
where \(C\) is a constant depending on \(n,\nu,\mu,\mu_1,R_{ij},R_3,R_4\).
Theorem 3. For any generalized solution \(u(x,t)\) from \(\mathring V_2^{1,0}\) of problem (1)—(2), the inequality
\[ |u|_{Q_T}\leq C\bigl[\|\psi_0\|_{L_2(D)} +\|\bar f\|_{L_2(Q_T)}+\|f\|_{L_{R_4}(Q_T)}\bigr] \]
holds, provided that assumptions (5)—(9) are fulfilled with respect to equation (1).
Theorem 4. If conditions (5)—(9) are fulfilled and \(\psi_0\in L_2(D)\), then problem (1), (2) has a solution from \(V_2(Q_T)\).
Theorem 5. Under conditions (5)—(9), any generalized solution \(u(x,t)\) of problem (1), (2) from \(\mathring V_2(Q_T)\) belongs to \(\mathring V_2^{1,1/2}(Q_T)\), and problem (1), (2) is uniquely solvable in \(\mathring V_2^{1,1/2}(Q_T)\), if \(\psi_0(x)\in L_2(D)\).
Theorem 6. If the coefficients of the equation satisfy conditions (5)—(9), then the boundary-value problem for (1), (2) cannot have two distinct generalized solutions from \(V_2(Q_T)\).
Theorem 7. Suppose \(u(x,t)\) is a generalized solution from \(\mathring V_2(Q_T)\) of problem (1), (2), and suppose \(a_{ij}, a_i\), and \(f_i\) satisfy conditions (5)—(9), while \(b_i(x,t), a(x,t)\), and \(f(x,t)\) satisfy the conditions
\[ \left\|\sum b_i^2,\; a\right\|_{L^{*}_{(r_1,r_2,l_1)}(Q_T)} \leq \mu_1,\qquad l_1\geq r_2\geq r_1;\qquad \|f\|_{L^{*}_{(r_{41},r_{42},l')}(Q_T)}\leq \mu_1; \]
\[ L_{(r_{11},r_2,l_1)}(Q_T)\in X_{(2-2/l_1)};\qquad L_{(r_{41},r_{42},l')}(Q_T)\in X_{((n+2)/2-1/l_1)}, \]
where
\[ L^{*}_{(m_1,m_2,\rho)}(Q_T)=L_{(\rho,m_2,m_1)}(0\leq t\leq T,\widetilde D_2,\widetilde D_1). \]
Then the function \(u(x,t)\) is an element of \(\mathring W_2^{1,-1/2}(Q_T)\).
Theorem 8. Suppose that for all operators
\[ \mathcal L^m u = u_t-\frac{\partial}{\partial x_i}\bigl(a_{ij}^m u_{x_j}+a_i^m u\bigr)+b_i^m u_{x_i}+a^m u,\qquad m=1,2,\ldots, \]
the conditions of Theorem 4 are satisfied with the same constants. Suppose that \(a_{ij}^m(x,t)\), remaining uniformly bounded, converge almost everywhere to \(a_{ij}\), and that the functions \(a_i^m, b_i^m, a^m, f_i^m, \psi_0^m\) converge to \(a_i, b_i, a, f, \psi_0\) in the norms of the spaces to which they belong under the conditions of Theorem 4. Then the generalized solutions \(u^m\) from \(V_2^{1,0}(Q_T)\) of the problems
\[ \mathcal L^m u \equiv \partial f_i^m/\partial x_i-f^m,\qquad u\big|_{S_T}=0,\qquad u\big|_{t=0}=\psi_0^m \]
converge strongly in \(V_2^{1,0}(Q_T)\) to the generalized solution \(u(x,t)\) of the limiting problem (1)—(2).
Theorem 9. Suppose that the coefficients of equation (1) satisfy the conditions
\[ \nu \xi_i\xi_i \leq a_{ij}\xi_i\xi_j \leq \mu \xi_i\xi_i,\qquad \nu,\mu=\mathrm{const}>0,\quad a_{ij}=a_{ji}, \]
\[ \|a,f,a_i^2,f_i^2\|_{L(r_1,r_2,l)(Q_T)}<\mu_1,\qquad \|b_i^2\|_{L(\bar r_1,\bar r_2,\bar l)(Q_T)}\leq \mu_1, \]
where \((r_1,r_2)\in \Omega_{(2-2/l_1)}^{(\varepsilon)}\), \((\bar r_1,\bar r_2)\in \Omega_{(2-2/\bar l_1)}\).
Then, for every generalized solution \(u(x,t)\) from \(V_2^{1,0}(Q_T)\) of equation (1), not exceeding \(k_0\) on \(\Gamma_T\), \(\operatorname*{vrai\,max} u(x,t)\) is finite and is estimated from above by a constant determined only by
\(n, k_0, \nu, \mu, \mu_1, r_1, r_2, l_1, \bar r_1, \bar r_2, \bar l_1\).
Theorem 10. Suppose that \(u(x,t)\) is a generalized solution from \(V_2^{1,0}(Q_T)\) of equation (1), whose coefficients \(a_{ij}, b_i, a\) satisfy the conditions
\[ \nu \xi_i\xi_i \leq a_{ij}\xi_i\xi_j \leq \mu \xi_i\xi_i,\qquad \nu,\mu=\mathrm{const}>0, \]
\[ \|b_i^2,a\|_{L(r_1,r_2,l)(Q_T)}\leq \mu_1,\qquad (r_1,r_1)\in \Omega_{(2-2/l_1)}, \]
\[ a(x,t)\geq 0,\qquad a_i=f_i=f=0. \]
Then for almost all \((x,t)\) in \(Q_T\)
\[ \min\left\{0,\operatorname*{vrai\,min}_{\Gamma_T} u(x,t)\right\} \leq u(x,t)\leq \max\left\{0,\operatorname*{vrai\,max}_{\Gamma_T} u(x,t)\right\}. \]
Theorem 11. Suppose that \(u(x,t)\) is a generalized solution from \(V_2^{1,0}(Q_T)\) of equation (1), whose coefficients and free terms satisfy the conditions of Theorem 9. Then, for any cylinder \(Q'\) lying at a positive distance \(d\) from \(\Gamma_T\), the quantity \(\operatorname*{vrai\,max}_{Q'} |u|\) is estimated from above by a constant depending only on
\(n,\nu,\mu,\mu_1,r_1,r_2,l_1,\bar r_1,\bar r_2,\bar l_1,d,\|u\|_{L_2(Q_T)}\).
Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Received
5 IX 1967
REFERENCES
- O. A. Ladyzhenskaya, V. A. Solonnikov, N. U. Ural’tseva, Linear and quasilinear equations of parabolic type, “Nauka,” 1967.
- O. A. Ladyzhenskaya, Tr. IV All-Union Mathematical Congress, 1, 1963.
- A. V. Ivanov, O. A. Ladyzhenskaya et al., Tr. Mat. Inst. im. V. A. Steklova, Academy of Sciences of the USSR, 92 (1965).
- O. A. Oleinik, Izv. AN SSSR, Ser. Mat., 25, 3 (1961).
- A. M. Il’in, A. O. Kalashnikov, O. A. Oleinik, UMN, 17, no. 3, 3 (1962).
- V. P. Il’in, Tr. Mat. Inst. im. V. A. Steklova, Academy of Sciences of the USSR, 66 (1962).