On solvability of nonclassical boundary value problems for differential equations of variable type (Q2713890)

The author studies the periodic problem for the second order parabolic equation with changing time direction NEWLINE\[NEWLINE k(t,x)u_t - Au + a(x,t)u=f(x,t),\;\;Au =\sum\limits_{i,j=1}^n (a_{ij}(x)u_{x_i})_{x_j} \;((t,x)\in Q=(0,T)\times \Omega). \tag{1} NEWLINE\]NEWLINE Here \(\Omega\subset {\mathbb R^n}\) is a bounded domain with smooth boundary and the operator \(A\) is uniformly elliptic. It is assumed that \(k(0,x)=k(T,x)\). Since the coefficient \(k\) may vanish, the periodic boundary condition is written as follows: NEWLINE\[NEWLINE u(0,x)|_{\Omega\setminus S_0}=u(T,x)|_{\Omega\setminus S_0},\;\;S_0=\{x\in \Omega: k(x,0)=0\}. NEWLINE\]NEWLINE The author considers either the Dirichlet or Neumann boundary condition on the lateral boundary of \(Q\). The main condition on the coefficients is the inequality \(2a(t,x)-|k_t(t,x)|\geq \delta>0\) (\((t,x)\in Q\)). Using this inequality, the smoothness assumptions for the coefficients and the right-hand side, and the Galerkin method, the author proves the existence and uniqueness of a solution \(u(t,x)\in L_2(0,T;W_2^2(\Omega))\cap W_2^1(Q)\).NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].