Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma

Abstract

In this paper we study several semilinear damped wave equations with "subcritical" nonlinearities, focusing on demonstrating lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. Under different assumptions imposed on the initial data, lifespan estimates from above are clearly showed. The key fact is that we find "transition surfaces", which distinguish lifespan estimates between "wave-like" and "heat-like" behaviours. Moreover we conjecture that the lifespan estimates on the "transition surfaces" can be logarithmically improved. As direct consequences, we reorganize the blow-up results and lifespan estimates for the massless case in which the "transition surfaces" degenerate to "transition curves". Furthermore, we obtain improved lifespan estimates in one space dimension, comparing to the known results.We also study semilinear wave equations with the scattering damping and negative mass term, and find that if the decay rate of the mass term equals to 2, the lifespan estimate is the same as one special case of the equations with the scaleinvariant damping and positive mass.The main strategy of the proof consists of a Kato's type lemma in integral form, which is established by iteration argument.We recall that the critical exponent p crit of (1.1) is the smallest exponent p crit > 1 such that, if p > p crit , there exists a unique global energy solution to the problem, whereas if 1 < p ≤ p crit the solution blows up in finite time. In the latter case, one is also interested in finding estimates for the lifespan T ε , which is the maximal existence time of the solution, depending on the parameter ε.Our principal model is the one in (1.1), for which we obtain Theorem 2 and Theorem 4, according to the different conditions imposed on the initial data. As straightforward consequences, we also obtain Theorem 1 and Theorem 3 for the massless case, i.e. the model with µ 2 = 0. The lifespan estimate in dimension n = 1 in this case is improved, comparing to the known results. Moreover, we continue the study of semilinear wave equations with scattering damping, negative mass term and power nonlinearity, introduced by the authors in [20,21].The paper is organized in this way: in the rest of the Introduction, we will sketch the background of the problems under consideration and we will exhibit our results, which will be proved in Section 3, exploiting, as main tool, a Kato's type lemma in integral form presented in Section 2.

Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture

“…Indeed, letting k = 0 in the definition of p c (k, n) above, then p c (0, 1) = +∞, and for n ≥ 2, p c (0, n) becomes the positive root of 2 + (n + 1)p − (n − 1)p 2 = 0, which is exactly the critical power for the small data Cauchy problem in (1.4) with k = 0(see e.g. [18] and references therein for background and results about this problem). Finally, we refer also to Ruan, Witt and Yin [27][28][29][30] for results about the local existence and local singularity structure of low regularity solutions for equation…”

A blow-up result for the wave equation with localized initial data: the scale-invariant damping and mass term with combined nonlinearities

“…), q S (N + µ)). We note that a recent improvement and a better comprehension of the transition from the heat-like equation to the wave-like one are obtained in [17]. On the other hand, a blow-up result is proven in [31] for all δ ≥ 0 and q ≤ q S (N + µ).…”