The critical exponent for semilinear σ-evolution equations with a strong non-effective damping

In this paper, we find the critical exponent for the existence of global small data solutions to: {u(tt) + (-Delta)(sigma)u + (-Delta)(theta/2) u(t) = f(u, u(t)), t >= 0, x epsilon R-n, (u, u(t))(0, x) = (0, u(1)(x)), in the case of so-called non-effective damping, theta epsilon (sigma, 2 sigma Sigma], where sigma not equal 1 and f = vertical bar u vertical bar(alpha) or f = vertical bar u(t)vertical bar(alpha), in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers alpha > (alpha) over tilde and do not exist, in general, for subcritical powers 1 < alpha < (alpha) over tilde. Assuming initial data to be small in L-1 or in some other Lp space, p epsilon (1, 2), in addition to the energy space, the critical exponent only depends on the ratio n/(Sigma p). We also prove the global existence of small data solutions in high space dimension for alpha > (alpha) over bar, but we leave open to determine if a counterpart nonexistence result for alpha < <(alpha)over tilde> holds or not. (C)2021 Published by Elsevier Ltd.

Automated summary

This paper investigates the critical exponent for the existence of global small data solutions in the case of non-effective damping, revealing the conditions for existence and non-existence of solutions for different exponents. The critical exponent depends only on the ratio of space dimension to p values.