Local well posedness of a 2D semilinear heat equation

Abstract

We investigate the initial value problem for a semilinear heat equation with exponential-growth nonlinearity in two space dimension. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space H 1 (R 2 ). The uniqueness part is non trivial although it follows Brezis-Cazenave's proof [3] in the case of monomial nonlinearity in dimension d ≥ 3. Next, we show that in the defocusing case our solution is bounded, and therefore exists for all time. In the focusing case, we prove that any solution with negative energy blows up in finite time. Lastly, we show that the unconditional result is lost once we slightly enlarge the Sobolev space H 1 (R 2 ). The proof consists in constructing a singular stationnary solution that will gain some regularity when it serves as initial data in the heat equation. The Orlicz space appears to be appropriate for this result since, in this case, the potential term is only an integrable function.2 is also the only one invariant under the same scaling (1.3). This property defines a sort of trichotomy in the dynamic of solutions of (1.2), and basically Date: October 29, 2018.

A note on global well-posedness and blow-up of some semilinear evolution equations

“…In the case of semilinear heat equation with exponential type nonlinearity, global well-posedness in some Orlicz space with small data is now proved [13]. Moreover, global well-posedness in the energy space holds for the defocusing sign [12].…”

“…3. Local well-posedness results in [10,6,12], are given for some explicit nonlinearities. The same arguments give local well-posedness for nonlinarities with the same behavior, which means the same growth at infinity.…”

Global well-posedness of a damped Schrödinger equation in two space dimensions

“…The fact that for any u ∈ H 1 and any a ≥ 0 (see ) implies that Applying Lebesgue's theorem, it follows that, when λ → 0 + , Because λ ↦ R ( λu 0 ) is continuous, there exists λ ∈ (0,1) such that R ( λu 0 ) = 0. So, L ( λu 0 ) ≥ m .…”

Local well‐posedness and blow‐up for an inhomogeneous nonlinear heat equation

“…Recently, the case of exponential nonlinearity was considered and results on global existence, blow‐up, and decay estimates were obtained (see, previous studies). See also Majdoub et al for the biharmonic heat equation.…”