We consider the Cauchy problem: {∂tu=Δu-u+λf(u)in(0,T)×R2,u(0,x)=u0(x)inR2,where λ> 0 , f(u):=2α0ueα0u2,for someα0>0,with initial data u∈ H1(R2). The nonlinear term f has a critical growth at infinity in the energy space H1(R2) in view of the Trudinger-Moser embedding. Our goal is to investigate from the initial data u∈ H1(R2) whether the solution blows up in finite time or the solution is global in time. For 0<12α0, we prove that for initial data with energies below or equal to the ground state level, the dichotomy between finite time blow-up and global existence can be determined by means of a potential well argument.

Asymptotics for a parabolic equation with critical exponential nonlinearity / M. Ishiwata, B. Ruf, F. Sani, E. Terraneo. - In: JOURNAL OF EVOLUTION EQUATIONS. - ISSN 1424-3199. - 21:2(2021), pp. 1677-1716. [10.1007/s00028-020-00649-z]

Asymptotics for a parabolic equation with critical exponential nonlinearity

B. Ruf;F. Sani;E. Terraneo
2021

Abstract

We consider the Cauchy problem: {∂tu=Δu-u+λf(u)in(0,T)×R2,u(0,x)=u0(x)inR2,where λ> 0 , f(u):=2α0ueα0u2,for someα0>0,with initial data u∈ H1(R2). The nonlinear term f has a critical growth at infinity in the energy space H1(R2) in view of the Trudinger-Moser embedding. Our goal is to investigate from the initial data u∈ H1(R2) whether the solution blows up in finite time or the solution is global in time. For 0<12α0, we prove that for initial data with energies below or equal to the ground state level, the dichotomy between finite time blow-up and global existence can be determined by means of a potential well argument.
Settore MAT/05 - Analisi Matematica
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2434/835028
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