Regularity and time behavior of the solutions to weak monotone parabolic equations

Abstract

In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

The Role of Data on the Regularity of Solutions to Some Evolution Equations

“…Proof of Corollary 2. The proof is similar at all to that of Corollary 1 once observed that if f ∈ L r (0, T; L σ (Ω)), with r and σ satisfying (30), by Theorem 1.3 in [8], it follows that the unique solution u 0 constructed by approximation of (11) satisfies ∇u 0 ∈ (L p (Ω T )) N .…”

“…We recall that if (2)-( 4) and ( 15) are satisfied for every T > 0, then there exists one and only one global solution constructed by approximation of (1) (see [30]). Theorem 1.…”