Non-Existence of Global Solutions for Higher-Order Evolution Inequalities in Unbounded Cone-Like Domains

Abstract

We use the test function method developed by Mitidieri and Pohozaev to get a priori estimates and non-existence results for semi-linear "higher-order evolution inequalities" in unbounded cone-like domains. As a model we consider the problem in a cone K with the positive initial-boundary conditionswhere ∆ denotes the Laplace operator.

Blow-up of solutions of a quasilinear parabolic equation

“…This lemma plays a crucial role in the proof of the proposition and thus theorem 1.1. The method of the proof of the lemma, as mentioned in § 1, relies on the test function method used in [22,23,25,26], which involves the judicious choice of a test function and the use of the Jensen inequality. However, we must develop it together with the Barenblatt solution (when m > 1).…”

“…As in [22,23,25], we consider ψ(x, t) = ψ R (x, t) q as a test function in the integral identity satisfied by u (see (2.1)). A simple calculation gives Then, setting…”

“…Our method relies on the test function method used in [22,23,25,26] (see also the references therein), which involves the judicious choice of a test function and the use of the Jensen inequality. This method is useful for showing the non-existence of global solutions to various problems (see, for example, [22,23,25]) and can be applied to our problem, albeit not directly. We must develop this method together with the Barenblatt solution (for m > 1) (see the proof of lemma 3.2).…”

Semilinear degenerate evolution inequalities with singular potential constructed from the generalized Greiner vector fields

“…Since Fujita's famous classical paper [4], the studies of existence and nonexistence of global solutions to nonlinear heat equations on the Euclidean space and on nilpotent Lie groups have attracted much interest over the past few years; see [5][6][7][8][9] and the references therein. Levine and Meier [6] and Pascucci [7] obtained some sharp critical exponents for the reaction-diffusion equation on the Euclidean space and on nilpotent Lie groups, respectively.…”

“…Levine and Meier [6] and Pascucci [7] obtained some sharp critical exponents for the reaction-diffusion equation on the Euclidean space and on nilpotent Lie groups, respectively. For the Euclidean case, Laptev [8] studied the nonexistence of global (nontrivial) solutions of some semilinear higher-order evolution inequalities. Hamidi and Laptev [9] proved nonexistence results for semilinear higher-order evolution inequalities with critical potential…”