Related papers: On inhomogeneous heat equation with inverse square…

We study Cauchy problem for the Hardy-H\'enon parabolic equation with an inverse square potential, namely, \[\partial_tu -\Delta u+a|x|^{-2} u= |x|^{\gamma} F_{\alpha}(u),\] where $a\ge-(\frac{d-2}{2})^2,$ $\gamma\in \mathbb R$, $\alpha>1$…

In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-square potential \[iu_{t} +\Delta u-c|x|^{-2}u=\lambda|x|^{-b} |u|^{\sigma } u,\; u(0)=u_{0} \in…

This work studies the Cauchy problem for the energy-critical inhomogeneous Hartree equation with inverse square potential $$i\partial_t u-\mathcal K_\lambda u=\pm |x|^{-\tau}|u|^{p-2}(I_\alpha *|\cdot|^{-\tau}|u|^p)u, \quad \mathcal…

We investigate the large time behavior of the hot spots of the solution to the Cauchy problem for the heat equation with a potential $\partial_t u-\Delta u+V(|x|)u=0$, where $V=V(r)$ decays quadratically as $r\to\infty$. In this paper,…

Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source $$ |x|^{-2}\partial_tu=\Delta u+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,T), $$ are…

In this paper, we study the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-power potential \[iu_{t} +\Delta u-c|x|^{-a}u=\pm |x|^{-b} |u|^{\sigma } u,\;\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where…

We consider the inhomogeneous nonlinear Schr\"odinger equation with inverse-square potential in $\mathbb{R}^N$ $$ i u_t + \mathcal{L}_a u+\lambda |x|^{-b}|u|^\alpha u = 0,\;\;\mathcal{L}_a=\Delta -\frac{a}{|x|^2}, $$ where $\lambda=\pm1$,…

In this paper we study the focusing inhomogeneous 3D nonlinear Schr\"odinger equation with inverse-square potential in the mass-supercritical and energy-subcritical regime. We first establish local well-posedness in $\dot{H}_a^{s_c}\cap…

In this paper we consider the initial value {problem $\partial_{t} u- \Delta u=f(u),$ $u(0)=u_0\in exp\,L^p(\mathbb{R}^N),$} where $p>1$ and $f : \mathbb{R}\to\mathbb{R}$ having an exponential growth at infinity with $f(0)=0.$ Under…

We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We…

Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the…

We investigate the Cauchy problem for a heat equation driven by the mixed local-nonlocal operator $\mathcal{L}:=-\Delta+(-\Delta)^s$, $s\in(0,1)$, with exponential nonlinearity \[ \partial_tu(x,t)+\mathcal{L}u(x,t)=f(u(x,t)), \qquad…

We study the Cauchy problem for the fractional semilinear heat equation with distributional inhomogeneous terms. By introducing the Lorentz--Morrey spaces, we overcome limitations of real interpolation in the classical local Morrey spaces…

We obtain $L^q$--$L^p$ decay estimates, $1\le q<p<\infty$ for solutions of nonlocal heat equations of the form $\partial_tu+\mathcal{L} u=0$. Here $\mathcal{L}$ is an integral operator given by a symmetric nonnegative kernel of L\'evy type.…

We study the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. For this aim, we establish decay estimates of the fractional heat semigroup in several uniformly local Zygumnd spaces.…

We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator $e^{-tH^{\beta}}$, $t, \beta>0$, associated with the harmonic oscillator $H=-\Delta + |x|^2$. We then prove some local and global…

In this paper, we consider the Cauchy global problem for the $L^2$-critical semilinear heat equations $\partial_t h=\Delta h\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In most…

We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey…

In this paper, we consider the Cauchy problem for the nonlinear Schr\"odinger equations with repulsive inverse-power potentials \[ i \partial_t u + \Delta u - c |x|^{-\sigma} u = \pm |u|^\alpha u, \quad c>0. \] We study the local and global…

Let $N\ge 1$ and let $f\in C[0,\infty)$ be a nonnegative nondecreasing function and $u_0$ be a possibly singular nonnegative initial function. We are concerned with existence and nonexistence of a local in time nonnegative solution in a…