Global existence of solutions for semi-linear wave equation with scale-invariant damping and mass in exponentially weighted spaces

Abstract

In this paper we consider the following Cauchy problem for the semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity: \begin{align}\label{CP abstract} \begin{cases} u_{tt}-Δu+\dfrac{μ_1}{1+t} u_t+\dfrac{μ_2^2}{(1+t)^2}u=|u|^p, \\ u(0,x)=u_0(x), \,\, u_t(0,x)=u_1(x), \end{cases}\tag{$\star$} \end{align} where ${\mu }_1,{\mu }_2^2$ are nonnegative constants and $p>1$. On the one hand we will prove a global (in time) existence result for \eqref{CP abstract} under suitable assumptions on the coefficients ${\mu }_1,{\mu }_2^2$ of the damping and the mass term and on the exponent $p$, assuming the smallness of data in exponentially weighted energy spaces. On the other hand a blow-up result for \eqref{CP abstract} is proved for values of $p$ below a certain threshold, provided that the data satisfy some integral sign conditions. Combining these results we find the critical exponent for \eqref{CP abstract} in all space dimensions under certain assumptions on ${\mu }_1$ and ${\mu }_2^2$. Moreover, since the global existence result is based on a contradiction argument, it will be shown firstly a local (in time) existence result.

Cited by in corpus (14)

• Fujita versus Strauss - a never ending story
• Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity
• Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma
• A global existence result for a semilinear wave equation with scale-invariant damping and mass in even space dimension
• The semilinear Euler-Poisson-Darboux equation: a case of wave with critical dissipation
• Critical exponent of Fujita-type for the semilinear damped wave equation on the Heisenberg group with power nonlinearity
• A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale-invariant lower order terms
• A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type
• Weakly coupled system of semilinear wave equations with distinct scale-invariant terms in the linear part
• Blow-up for wave equation with the scale-invariant damping and combined nonlinearities
• Wave-like blow-up for semilinear wave equations with scattering damping and negative mass term
• Integral representation formulae for the solution of a wave equation with time-dependent damping and mass in the scale-invariant case
• Short time blow-up by negative mass term for semilinear wave equations with small data and scattering damping
• The move from Fujita to Kato type exponent for a class of semilinear evolution equations with time-dependent damping