Local and global time decay for parabolic equations with super linear first-order terms

Abstract

We study a class of parabolic equations having first-order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton-Jacobi equation with superlinear Hamiltonian. We address the problem of having unbounded initial data and we develop a local theory yielding well-posedness for initial data in the optimal Lebesgue space, depending on the superlinear growth. Then we prove regularizing effects, short and long time decay estimates of the solutions. Compared to previous works, the main novelty is that our results apply to nonlinear operators with just measurable and bounded coefficients, since we totally avoid the use of gradient estimates of higher order. By contrast we only rely on elementary arguments using equi-integrability, contraction principles and truncation methods for weak solutions.

Regularizing effect and decay results for a parabolic problem with repulsive superlinear first order terms

“…PROOF. We omit the proof since it is very similar to the one proposed in [18,Lemma 3.1]. We just observe that the growth assumption (3.8) plays the role of (RC).…”

“…Roughly speaking, we prove that a contraction in the L σnorm, σ > 1 as in (ID σ ), holds for the level set function G k (u(t)) provided that this is initially (t = 0) not too big (i.e., k is large). We underline that, when dealing with the G k (•) function, no smallness conditions on the initial datum are assumed, but eventually it is enough to take a large k. An analogous δ argument has already been used in [18] where (P) is studied under the assumptions in SECTION 2 when p = 2.…”

“…A nonlinear approach, aimed at studying the regularity and the behaviour in time of solutions of (1.1) with p = 2, is contained in [18]. In particular, the main step relies on the proof of an a priori estimate for the level set function G k (u) = (|u| − k) + sign(u) which has the form sup t∈(0,T) G k (u(t))…”

“…Dealing with a general superlinear setting, then one has to require some regularity on the solutions in order to have the problem well posed. In this sense, we refer to [2,25] in the elliptic framework and [18,16] in the parabolic one. More precisely, a comparison result is proved in [18,Section 6] when the solution u belongs to the regularity class u solving (1.1) : |u|…”

“…In this sense, we refer to [2,25] in the elliptic framework and [18,16] in the parabolic one. More precisely, a comparison result is proved in [18,Section 6] when the solution u belongs to the regularity class u solving (1.1) : |u|…”