Estimates of the highest-order derivatives in some nonlinear parabolic equations (Q1814390)

The present paper studies second order fully nonlinear parabolic equations in one space variable. Proofs of existence of solutions of mixed problems for such equations are based, as a rule, on obtaining a priori estimates in the corresponding function spaces. Apart from certain smoothness requirements on the coefficients and conditions of compatibility of the initial data with the boundary conditions, we impose a number of restrictions on the problem itself (the nature of the nonlinearity). Some of these allow us to obtain a priori estimates for the solution in the space \(C\), while others guarantee boundedness of the derivative \(u_ x (x,t)\) once we have an estimate of the maximum of the modulus of the solution itself. For a quasilinear equation of the form \(u_ t= a(x,u, u_ x) u_{xx}+ b(x,u, u_ x)\) this estimate of the derivative with respect to the spatial variable is essential in the sense that once it is obtained, one can obtain, without any additional restrictions on the structure of the problem, an estimate of the solution in the space \(C_{xt}^{2+ \alpha, 1+\alpha/2}\), \(\alpha>0\) (if the coefficients are sufficiently smooth). However, in the case of a nonlinear equation of a more general form, \(u_ t+ a (t,x, u, u_ x, u_{xx})\), to use this procedure (consecutively obtaining estimates of the solution in stronger and stronger norms, up to Hölder norms of the highest order derivatives), we need to establish boundedness of \(| u_{xx}|\). In the present paper, we obtain an estimate of \(| u_{xx} |\) after having established boundedness of \(\| u\|_{C_{x,t}^{ 1,0}}\). We propose to use a second order Lyapunov functional (the order of a functional we define to be the order of the highest derivative used in its definition).