Existence and uniqueness of solutions for a class of doubly degenerate parabolic equations

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.…”

“…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].…”

Doubly Degenerate Parabolic Equation with Time-Dependent Gradient Source and Initial Data Measures

“…erefore, (8)-(11) follow from (53), (20), (26), (28), and (29), provided that we can indeed find a quantitative estimates below t 3 ≥ T 0 . We may assume t 3 < T * , since the estimate is otherwise trivial.…”

“…However, the existence of solutions for initial data measures was left open. In the bounded domain, the critical extinction exponent, the existence, and uniqueness of weak solutions were studied [28,29]. Weng [30] established the existence and stability of weak solutions to the double degenerate evolutionary p(x)-Laplacian equation.…”

“…Proof. In the following, we only prove (28) and (30), since (29) and (31) can be proved similarly and we omit the details.…”