The local estimates for solutions of the elliptic-parabolic type equations in the space of S. L. Sobolev (Q2742849)

Let \(D(r)\) be a ball or a semi-ball of the radius \(r\) or a cylinder in \(\mathbb R^n\). The following equation NEWLINE\[NEWLINEK(x,t) u_{tt}+\sum_{i,j=1}^na_{i,j}(x,t)u_{x_ix_j}+\sum_{i=1}^nb_i(x,t)u_{x_i}+a(x, t)u_t+c(x,t)u=f(x,t),NEWLINE\]NEWLINE \(x\in D(r)\), \(t\in(0,T)\), where \(a_1 \xi ^2\leq\sum_{i,j=1}^na_{i,j}(x,t) \xi_i\xi_j\leq a_2 \xi ^2\) in \(\overline D(r)\) for all \(\xi\in\mathbb R^n\) and \(a_{i,j}=a_{j,i}\), \(a_1,a_2\) -- positive numbers, \(K\geq 0\) in \(\overline D(r)\times[0,T]\), \(K(x,0)=K(x,T)=0\) and \(a(x,0)-K_t(x,0)=a(x,T)-K_t(x,T)\not=0\) for \(x\in D(r)\) is considered. A local a priori estimate of generalized solutions to the above equation satisfying the conditions \(u(x,T)= \lambda u(x,0) (\lambda\not=0)\) for \(x\in D(r)\) and \(u=0\) on \(\Gamma= \gamma\times(0,T)\), where \(\gamma=\{0\}\) if \(D(r)\) is a ball and, if \(D(r)\) is a cylinder, then \(\gamma\) is its lateral surface, is obtained.