Solutions of nonlinear parabolic equations without growth restrictions on the data (Q2737489)

The paper deals with solutions in \(\mathbb{R}^N\) to nonlinear parabolic equations of the form \(u_t+L(u)+h(x,t,u)=0\), with initial condition \(u(x,0)=u_0(x)\). \(L(u)\) is an elliptic differential operator in divergence form with some structure conditions, which include the standard \(p\)-Laplacian operator, and \(h(x,t,u)\) is a function which grows uniformly with \(u\) at a sufficient rate. If \(L(u)\) is the \(p\)-Laplacian then the rate of growth of \(h\) is greather than \(p-1\). The function \(u_0(x)\) is assumed to be locally integrable in \(\mathbb{R}^N\), without any control of its growth at infinite. First, the authors establish a priori estimates of local type for a sequence of suitable approximate problems, then, passing to the limit and using the conditions in above, they prove existence of a solution for the initial problem. The corresponding results for elliptic problems were already studied. Also the regularity of the solution is investigated. In particular, for large rates of growth of \(h\) with respect to \(u\), the regularity of \(u\) and \(Du\) is improved with respect to the results known from the standard theory. Finally, the question of uniqueness of local solutions is discussed.