A Constructive Existence Theorem for a Nonlinear Parabolic Equation

Abstract

This paper considers the first boundary value problem for the quasilinear parabolic differential equation \[({\text {A}})\qquad L[u] = \sum_{i,j = 1}^n {a_{ij} (x,t)u_{x_i x_j } + } \sum_{i = 1}^n {b_i (x,t)u_{x_i } - u_t = - f(x,t,u)}\] on a cylindrical domain in $E_{n + 1} $. The central theorem is an existence theorem offering an iterative procedure for approximating or constructing a classical solution $u(x,t)$ for this problem. The iteration is done by successively applying a monotone integral operator with an appropriate Green’s function as its kernel. The procedure starts with either of a pair of functions $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma } (x,t) \leqq \bar \sigma (x,t)$ which satisfy \[L[\bar \sigma ] \geqq - f(x,t,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\sigma } )\] and \[L[\bar \sigma ] \leqq - f(x,t,\bar \sigma ).\] This theorem is then used to prove an existence theorem for equation (A) on $E_{n + 1} $ by solving a sequence of problems on nested cylinders in $E_{n + 1} $.