A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations

In this paper, we find the critical exponent for global small data solutions to the Cauchy problem in  Rn, for dissipative evolution equations with power nonlinearities  |u|p or  |ut|p,utt+(−Δ)δut+(−Δ)σu=|u|p,|ut|p. Here  σ,δ∈N∖0, with  2δ≤σ. We show that the critical exponent for each of the two nonlinearities is related to each of the two possible asymptotic profiles of the linear part of the equation, which are described by the diffusion equations: vt+(−Δ)σ−δv=0,wt+(−Δ)δw=0. The nonexistence of global solutions in the critical and subcritical cases is proved by using the test function method (under suitable sign assumptions on the initial data), and lifespan estimates are obtained. By assuming small initial data in Sobolev spaces, we prove the existence of global solutions in the supercritical case, up to some maximum space dimension  n̄, and we derive  Lq estimates for the solution, for  q∈(1,∞). For  σ=2δ, the result holds in any space dimension  n≥1. The existence result also remains valid if  σ and/or  δ are fractional

Asymptotic stability and blow up for a semilinear damped wave equation with dynamic boundary conditions

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained

On the Unsolvability Conditions for Quasilinear Pseudohyperbolic Equations

In this paper, we study the nonexistence of global weak solutions to the Cauchy problem of quasilinear pseudohyperbolic equations with damping term. The sufficient conditions for nonexistence of nontrivial global weak solutions is obtained in terms of exponents, singularities order and other parameters in the problem. The nonlinear capacity method is applied to prove nonexistence theorems. The proofs of our nonexistence theorems are based on deriving apriori estimates for the possible solutions to the problem by an algebraic analysis of the integral form of inequalities with an optimal choice of test functions. The result is extended to the case of coupled system of quasilinear pseudohyperbolic equations

The critical exponent for an ordinary fractional differential problem

AbstractWe consider the Cauchy problem for an ordinary fractional differential inequality with a polynomial nonlinearity with variable coefficient. A nonexistence result is proved and the critical exponent separating existence from nonexistence is found. This is proved in the absence of any regularity assumptions