Asymptotics for a parabolic equation with critical exponential nonlinearity
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Journal of Evolution Equations
Asymptotics for a parabolic equation with critical exponential nonlinearity
Michinori Ishiwata, Bernhard Ruf, Federica Sani and Elide Terraneo
Abstract. We consider the Cauchy problem:
∂t u = u − u + λ f (u) in (0, T ) × R2 , u(0, x) = u 0 (x) in R2 ,
where λ > 0, 2
f (u):=2α0 ueα0 u , for some α0 > 0, with initial data u 0 ∈ H 1 (R2 ). The nonlinear term f has a critical growth at infinity in the energy space H 1 (R2 ) in view of the Trudinger-Moser embedding. Our goal is to investigate from the initial data u 0 ∈ H 1 (R2 ) whether the solution blows up in finite time or the solution is global in time. For 0 < λ < 2α1 , 0 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.
1. Introduction and main results Model parabolic problem. We consider the Cauchy problem for a two-space dimensional parabolic equation with exponential-type nonlinearity; more precisely, we focus the attention on the following model problem:
∂t u = u − u + λ f (u) in (0, T ) × R2 , u(0, x) = u 0 (x)
in R2 ,
where λ > 0, f (u) := 2α0 ueα0 u , for some α0 > 0, 2
and we consider initial data in the energy space H 1 (R2 ), i.e., u 0 ∈ H 1 (R2 ).
(1.1)
J. Evol. Equ.
M. Ishiwata et al.
In this framework, energy refers to the functional associated with the stationary problem: 1 2 F(v) dx, I (v) := v H 1 − λ 2 R2 where v H 1 :=
∇v2L 2
+ v2L 2
1 2
v
, and F(v) :=
f (η) dη = eα0 v − 1. 2
0
The above functional is well defined in H 1 (R2 ), and the nonlinear term f that we are considering has critical growth in the energy space in view of the Trudinger–Moser embedding [1,32]. Concerning local existence and uniqueness for (1.1), Ibrahim, Jrad, Majdoub and Saanouni [14] proved that, for any u 0 ∈ H 1 (R2 ), the Cauchy problem (1.1) has a local in time solution u ∈ C [0, T ]; H 1 (R2 ) for some finite time T > 0 (see Definition 2.1 and Remark 2.2), and the solution is unique. Then, the smoothing effect of the heat kernel implies that u is a classical solution; in fact, it belongs to the class ∞ (0, T ]; L ∞ (R2 ) ∩ C 1 (0, T ); L 2 (R2 ) ∩ C 1,2 (0, T ) × R2 , u ∈ L loc see [20, Remark 4.1]. We define the maximal existence time T∗ of the solution u as
T∗ := sup T > 0 : the problem (1.1) admits a solution u ∈ C [0, T ]; H 1 (R2 ) ∈ (0, +∞].
If T∗ < +∞, then the L ∞ -norm of the solution blows up, i.e., if T∗ < +∞ then lim sup u(t) L ∞ = +∞, t→T∗
see, e.g., [5, Section 5.3]. In view of the definition of T∗ , it is natural to try to understand whether T∗ < +∞ yields also the blow-up of the H 1 -norm of the solution. This problem is related to the dependence of the local existence time of the solution to (1.1) from the size of the initial data u 0 ∈ H 1 (R2 ); this aspect is emphasized in Sect. 2 in comparison with the energy subcritical problem. For the energy subcritical problem, the local existence time
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