Asymptotics for a class of heat equations with inhomogeneous nonlinearity

We study some energy well-posedness issues of the Schrödinger equation with an inhomogeneous mixed nonlinearity and radial data $$ i\dot{u}-(-\Delta )^{s} u \pm \vert x \vert ^{\rho} \vert u \vert ^{p-1}u\pm \vert u \vert ^{q-1}u=0, \quad 0< s< 1, \rho \neq 0, p,q>1. $$ i u ˙ − ( − Δ ) s u ± | x | ρ | u | p − 1 u ± | u | q − 1 u = 0 , 0 < s < 1 , ρ ≠ 0 , p , q > 1 . Our aim is to treat the competition between the homogeneous term $|u|^{q-1}u$ | u | q − 1 u and the inhomogeneous one $|x|^{\rho}|u|^{p-1}u$ | x | ρ | u | p − 1 u . We simultaneously treat two different regimes, $\rho >0$ ρ > 0 and $\rho <-2s$ ρ < − 2 s . We deal with three technical challenges at the same time: the absence of a scaling invariance, the presence of the singular decaying term $|\cdot |^{\rho}$ | ⋅ | ρ , and the nonlocality of the fractional differential operator $(-\Delta )^{s}$ ( − Δ ) s . We give some sufficient conditions on the datum and the parameters N, s, ρ, p, q to have the global versus nonglobal existence of energy solutions. We use the associated ground states and some sharp Gagliardo–Nirenberg inequalities. Moreover, we investigate the $L^{2}$ L 2 concentration of the mass-critical blowing-up solutions. Finally, in the attractive regime, we prove the scattering of energy global solutions. Since there is a loss of regularity in Strichartz estimates for the fractional Schrödinger problem with nonradial data, in this work, we assume that $u_{|t=0}$ u | t = 0 is spherically symmetric. The blowup results use ideas of the pioneering work by Boulenger el al. (J. Funct. Anal. 271:2569–2603, 2016).

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Asymptotics for a class of heat equations with inhomogeneous nonlinearity

Abstract: The initial value problem for an inhomogeneous nonlinear heat equation with a pure power nonlinearity is investigated. In the radial energy space, global and non-global well-posedness are discussed.

Cited by 2 publications

References 20 publications

Scattering for a Radial Defocusing Inhomogeneous Choquard Equation

Scattering for a Radial Defocusing Inhomogeneous Choquard Equation

A note on inhomogeneous fractional Schrödinger equations

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