First boundary value problem for Cordes-type semilinear parabolic equation with discontinuous coefficients (Q2194645)

Summary: For a class of semilinear parabolic equations with discontinuous coefficients, the strong solvability of the Dirichlet problem is studied in this paper. The problem \[\sum_{i,j =1}^n a_{ij} (t,x) u_{x_i x_j} - u_t + g (t,x,u)=f (t,x), \quad u|_{\Gamma (Q_T)}=0,\] in \(Q_T=\Omega\times (0,T)\) is the subject of our study, where \(\Omega\) is bounded \(C^2\) or a convex subdomain of \(E_{n+1}\), \(\Gamma (Q_T)=\partial Q_T\backslash \{t=T\}\). The function \(g(x,u)\) is assumed to be a Caratheodory function satisfying the growth condition \(|g (t,x,u)| \leq b_0 |u|^q\), for \(b_0>0\), \(q\in (0,(n+1) / (n-1))\), \(n\geq2\), and leading coefficients satisfy Cordes condition \(b_0>0\), \(q\in (0,(n+1) / (n-1))\), \(n\geq 2\).