On solvability of some boundary value problems for a class of quasilinear parabolic equations
The article is devoted to studying the solvability of initial-boundary value problems for the nonlinear parabolic equation \[ u_t =a(t,x,u,u_x)u_{xx}+ F(t,x,u,u_x),\;\;(t,x)\in Q=(0,T)\times (-l,l)\;(l>0). \tag{1} \] Equation (1) is furnished with the boundary condition of the second, or third boundary value problem, or the problem when on one end of the interval \((-l,l)\) these conditions are replaced by the Dirichlet boundary condition. The first question considered in the article is a priori bounds for the maximum of the function \(u_x\) under the condition that a classical solution \(u\) is bounded. The second question are existence and uniqueness theorems of a classical solution to the above-mentioned problems. The peculiarities of the results are in the fact that the growth condition for the function \(F\) does not coincide with the celebrated Bernshteĭn condition. It is assumed that the function \(F\) is representable as \[ F(t,x,u,p)=b(t,x,u,p)+ c(t,x,p) \] or \[ F(t,x,u,p)=b(t,x,u,p)+ c(t,u,p), \] where the function \(b/a\) meets the Bernshteĭn condition (i.e., the growth in \(p\) as \(|p|\to \infty\) is at most 2), and \(c\) is of arbitrary growth in \(p\) and satisfies some simple additional conditions; in particular, there are no conditions at all if \(c=c(t,p)\).
existence and uniqueness theorems of a classical solution
scientific article; zbMATH DE number 1396679