Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds

Abstract

Abstract. We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.

A Liouville Theorem for Superlinear Heat Equations on Riemannian Manifolds

“…for p ≤ 1 + 2/n), in accordance with [21] and [22]. Moreover, we underline that previous similar Liouville-type theorems (see for instance [14] and references therein) require the nonnegativity of the solutions, while here we just assume their boundedness from below.…”

“…Subsequently, several generalizations to more quasilinear parabolic operators (p-Laplacian or porous medium-type, for instance) have been developed, see [8,9,16,18] and references therein. At the same time, extensions have been carried out also in the context of Riemannian manifolds: in the papers [21,22] (see also [14] and references therein), Zhang proved the same result of Fujita for the equation u t = ∆u + |u| p V on a Riemannian manifold (M, g), under some geometrical assumptions and growth conditions on the weight V in the case this latter is time-independent. In this work we remove the positivity hypothesis by such conclusion of nonexistence of nonzero solutions (equivalently, of triviality), assuming only a bound from below that, as we will see, implies the positivity of the solutions, moreover, we extend the work of Zhang by considering weights V which can depend also on time.…”

Porous medium equations on manifolds with critical negative curvature: unbounded initial data

“…Recently, quasilinear degenerate (or singular) parabolic equations of the type of (1.1) on Riemannian manifolds have attracted a lot of interest: with no claim to completeness, we quote e.g. [26,19,20,22,23,5,14,25,6,9,13,7,8] (see also the significant papers [15,10,12,11,16] in the purely parabolic case). In particular, in [7] problem (1.1) has thoroughly been studied under the assumption In [7,Theorem 2.2] it is shown that if the initial datum complies with the growth estimate |u 0 (x)| ≤ C (1 + ρ(x)) σ m−1 for a.e.…”

On the existence of nonnegative solutions to semilinear differential inequality on Riemannian manifolds

“…Subtituding above into (15) , we have But u 0 ≥ 0 is not identically zero, by Maximum principle, we know that u is positive almost everywhere, which leads a contradiction, and then there's no nonnegative solution to problem (1).…”

“…Recently, this idea was used and developed for solutions to both elliptic equations (see, e.g. [11,12,18,19]) and parabolic equations (see [15]). Particularly in [15], Mastrolia, Monticelli and Punzo proved that if…”