Improvement on the blow-up of the wave equation with the scale-invariant\n damping and combined nonlinearities

2020/08/03 by Makram Hamouda, Hamouda, Makram, Mohamed Ali Hamza +1 · 2 citations
Engineering · Mathematics · #35B44 #35L71 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2008.02109

openalex publication_date 2020/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider in this article the damped wave equation, in the\n\scale-invariant case with combined two nonlinearities, which reads as\nfollows: \
d (E)
hspace1cm u_tt-
Deltaν+
frac
mu1+tu_t=|u_t|^p+|u|^q,
quad
mboxin

R^N
times[0,
infty),\n with small initial data. Compared to our previous work\n citeOur, we show in this article that the first hypothesis on the damping\ncoefficient \μ, namely \μ < \(N(q-1))/(2), can be removed, and the\nsecond one can be extended from (0, \μ_*/2) to (0, \μ_*) where \μ_*>0\nis solution of (q-1)\((N+\μ_*-1)p-2\) = 4. Indeed, owing to a\nbetter understanding of the influence of the damping term in the global\ndynamics of the solution, we think that this new interval for \μ describe\nbetter the threshold between the blow-up and the global existence regions.\nMoreover, taking advantage of the techniques employed in the problem (E), we\nalso improve the result in citeLT2,Palmieri in relationship with the Glassey\nconjecture for the solution of (E) but without the nonlinear term |u|^q.\nMore precisely, we extend the blow-up region from p \∈ (1, p_G(N+\σ)],\nwhere \σ is given by eqrefsigma below, to p \∈ (1, p_G(N+\μ)]\ngiving thus a better estimate of the lifespan in this case.\n

Cited by

• Blow-up results for a Nakao-type problem with a time-dependent damping term and derivative-type nonlinearities
• On the blow-up of solutions to a Nakao-type problem with a time-dependent damping term

Related

• A blow-up result for the wave equation with localized initial data: the scale-invariant damping and mass term with combined nonlinearities
• Improvement on the blow-up for a weakly coupled wave equations with scale-invariant damping and mass and time derivative nonlinearity
• New blow-up result for the weakly coupled wave equations with a scale-invariant damping and time derivative nonlinearity
• Blow-up and lifespan estimate for wave equations with critical damping term of space-dependent type related to Glassey conjecture
• A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent