Comparison results for unbounded solutions for a parabolic Cauchy-Dirichlet problem with superlinear gradient growth

Abstract

In this paper we deal with uniqueness of solutions to the following problemthe parabolic cylinder, Ω is an open subset of R N , N ≥ 2, 1 < p < N , and the right hand side H(t, x, ξ) : (0, T ) × Ω × R N → R exhibits a superlinear growth with respect to the gradient term. 2010 Mathematics Subject Classification. 35B51, 35K55,35K61.

Lipschitz regularity for viscous Hamilton-Jacobi equations with Lp terms

“…Note that when Q = ∞ and P = d one has to assume an additional hypothesis on B, namely that |B(·, t)| d is uniformly integrable with respect to t ∈ (0, T ). We mention that, at least in the sub-quadratic case γ < 2, it is known that uniqueness holds under weaker conditions on u involving suitable powers of u itself in L 2 (0, T ; W 1,2 ), see [18] and references therein.…”

On some nonlocal parabolic systems with gradient source terms

“…Complete classification of the set of nonnegative solutions was obtained in relation with the classical parabolic capacity (see [5]). Concerning the case of unbounded domains strictly included in R N , we refer to [53] and references cited therein. -Nonlocal case 1 2 < s < 1.…”

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

“…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|…”