Narrow Results

We consider a quasilinear equation (see (1.1)) with L¹ data and with a diffusion matrix A(x,u), which is not uniformly coercive with respect to u (see Assumptions (H3)–(H4)). Under such assumptions it is not realistic, in general, to search for a solution which is finite almost everywhere. We introduce two equivalent notions of solutions which take into account the possible values +∞ and -∞ (see Definitions 2.1 and 2.3). Then we prove that there exists at least one such solution. At last we establish an uniqueness result in the class of simultaneous infinite valued solutions.

In this paper, we prove, the existence of a renormalized solution for a class of nonlinear parabolic problems whose prototype is