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.
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.
It is the purpose of this work to obtain a sharp threshold of global existence vs blow-up dichotomy for mass-super-critical and energy subcritical solutions to an inhomogeneous Choquard equation.
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