Spatial decay theorems for nonlinear parabolic equations in semi-infinite cylinders

Abstract

Classes of nonlinear parabolic equations in a semi-infinite cylinder are considered. The equations are of the form

$$u,_{jj} + g_{ij} \left( {x,u,p,\partial ^2 u} \right)u,_i u,_j = c\left( {x,u,p} \right)\frac{{\partial u}}{{\partial t}},$$

wherep=u, _k u, _k and∂ ² u represents a general space derivative of second order. Homogeneous Dirichlet data are prescribed on the lateral sides of the cylinder for all time, along with zero initial data. At any fixed timet, the solution is assumed to be bounded throughout the cylinder, as is the corresponding symmetric matrixg _ij . Under these assumptions, it is proved that each solution decays pointwise exponentially to zero with distance from the face of the cylinder and the exponential decay rate depends only upon the cross-section of the cylinder, but not upon time or the bounds foru andg _ij . In addition, if the boundary data on the face of the cylinder satisfy certain mild smoothness conditions, one obtains a decay rate equal to the best possible rate for the Laplace equation.