Existence of solutions for a class of porous medium type equations with data

Existence of solutions for some doubly degenerate parabolic equations with natural growth terms

“…Inspired by the ideas in [1,13,25,20,32], we shall prove the existence of bounded solutions to (P) in the standard weak sense, that is the solution belongs to L p (0, T ; W 1,p 0 (Ω )) ∩ L ∞ (Q), which improves the previous results [1,20,33,25] in some senses. There exist two main difficulties in dealing with this problem, which are related to the facts that b is only assumed to be a strictly C 1 function and the equation is doubly degenerate.…”

“…The proof of Theorem 3.1 is similar to the proof of Theorem 1 in [7] (see also [32]). However, in order to have a self-contained paper, let us briefly explain the argument.…”

“…In the case that b(u) = u and α(0) = 0, existence results are proved in [20,32], existence and uniqueness results are shown in [33] with F = 0, and existence results to the stationary case of (P) can be found in [25,31]. We point out that the distributional solution u obtained in [20] for p = 2, may not belong to L 2 (0, T ; H 1 0 (Ω )), that is, to define ∇u almost everywhere in Q, one may either use the notion of a generalized gradient as in [3] or employ the way as in (2.8) of [6].…”