Doubly Nonlinear Equations of Porous Medium Type

Abstract

In this paper we prove the existence of solutions to doubly nonlinear equations whose prototype is given by

$$\partial_t u^m- {\rm div}\, D_{\xi}\, f(x,Du) =0,$$

with ${m\in (0,\infty )}$ , or more generally with an increasing and piecewise C¹ nonlinearity b and a function f depending on u

$$\partial_{t}b(u) - {\rm div}\, D_{\xi}\, f(x,u,Du)= -D_u f(x,u,Du).$$

For the function f we merely assume convexity and coercivity, so that, for instance, ${f(x,u,\xi)=\alpha(x)|\xi|^p + \beta(x)|\xi|^q}$ with 1 < p < q and non-negative coefficients α, β with ${\alpha(x)+\beta(x)\geqq \nu > 0}$ , and ${f(\xi)=\exp(\tfrac12|\xi|^2)}$ are covered. Thus, for functions ${f(x,u,\xi )}$ satisfying only a coercivity assumption from below but very general growth conditions from above, we prove the existence of variational solutions; mean while, if ${f(x,u,\xi )}$ grows naturally when ${\left\vert \xi \right\vert \rightarrow +\infty }$ as a polynomial of order p (for instance in the case of the p-Laplacian operator), then we obtain the existence of solutions in the sense of distributions as well as the existence of weak solutions. Our technique is purely variational and we treat both the cases of bounded and unbounded domains. We introduce a nonlinear version of the minimizing movement approach that could also be useful for the numerics of doubly nonlinear equations.