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  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.
Abstract. Using a sharp Gagliardo-Nirenberg type inequality, well-posedness issues of the initial value problem for a fractional inhomogeneous Schrödinger equation are investigated. Consider the initial value problem for an inhomogeneous nonlinear Schrödinger equation
Contentswhich models various physical contexts in the description of nonlinear waves such as propagation of a laser beam and plasma waves. For example, when γ = 0, it arises in nonlinear optics, plasma physics and fluid mechanics [2,3]. When γ > 0, it can be thought of as Date: February 16, 2016. 1991 Mathematics Subject Classification. 35Q55.
We consider the initial value problem for a two-dimensional semi-linear wave equation with exponential type nonlinearity. We obtain global well-posedness in the energy space. We also establish the linearization of bounded energy solutions in the spirit of Gérard . The proof uses Moser-Trudinger type inequalities and the energy estimate.
We investigate the initial value problem for a semi-linear fractional damped Schrödinger equation. Global existence and scattering are proved depending on the size of the damping coefficient. C 2015 AIP Publishing LLC.
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