Related papers: On a non-homogeneous and non-linear heat equation

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…

We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first…

We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation $i \partial_{t}u+ \Delta u=\lambda_{0}u+\lambda_{1}|u|^\alpha u$ in $\mathbb{R}^{N}$, where $\lambda_{0},\lambda_{1}\in\mathbb{C}$, in $H^s$ subcritical and critical…

The aim of the paper is to study the problem $$ \begin{cases} u_{tt}-\Delta u+P(x,u_t)=f(x,u) \qquad &\text{in $(0,\infty)\times\Omega$,} u=0 &\text{on $(0,\infty)\times \Gamma_0$,} u_{tt}+\partial_\nu u-\Delta_\Gamma…

The global dynamics and regularity of parabolic-hyperbolic systems is an interesting topic in PDEs due to the coupling of competing dissipation and hyperbolic effects. This paper is concerned with the Cauchy problem of a…

The first part of this paper is devoted to the derivation of a technical result, related to the stability of the solution of a reaction-diffusion equation $u_t-\Delta u = f(x,u)$ on $(0,\infty)\times \mathbb{R}^N$, where the initial datum…

In this article, we consider the Cauchy problem for the cubic (mass-critical) Zakharov-Kuznetsov equations in dimension two: $$\partial_t u+\partial_{x_1}(\Delta u+u^3)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}^{2}.$$ For initial data in…

The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem $\partial_t u = \Delta_p u + |\nabla u|^q$ when the initial data converge to zero at infinity. Sufficient conditions on…

In this paper we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schr\"odinger equation $i\partial_{t}u+\Delta u=\lambda|x|^{-\alpha}|u|^{\beta}u$ in $H^1$. The well-posedness theory in $H^1$ has been intensively…

We consider the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ^2) u= \pm \partial (|u|^2u)$ on $\mathbb{R} ^d$, $d \ge 3$, with random initial data, where…

We consider the Cauchy problem for a time fractional semilinear heat equation with initial data belonging to inhomogeneous/homogeneous Besov--Morrey spaces. We present sufficient conditions for the existence of local/global-in-time…

In this paper, we consider the Cauchy problem for a semilinear damped wave equation with the nonlinear term $|u|^{1+2/n} \mu(|u|)$, where $\mu$ is a modulus of continuity. In recent papers by Ebert,Girardi,Reissig (Math. Ann. 378 (2020))…

We study the Cauchy problem for the fractional Schr\"{o}dinger equation $$ i\partial_tu = (m^2-\Delta)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < \alpha < 2$, and $F$ stands for the nonlinearity of…

Consider the equation u_t=\Delta u-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=\phi(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\frac\omega{|x|^2}$ as $|x|\to\infty$, for some $\omega\neq0$, and $a(x)$ is on the…

We consider the nonlinear heat equations with Neumann boundary conditions $$ \begin{cases} u_{t}=\Delta u & \text{in}\ \mathbb{R}_{+}^{4} \times(0, T) ,\\ -\frac{d u}{d x_{4}}(\tilde{x}, 0, t) \ =u^2(\tilde{x}, 0, t)& \text{in}\…

We study the behavior as $t\to 0^+$ of nonnegative functions \begin{equation}\label{0.1} u\in C^{2,1} (\mathbb{R}^n\times (0,1)) \cap L^\lambda (\mathbb{R}^n\times (0,1)),\quad n\ge 1, \end{equation} satisfying the parabolic Choquard-Pekar…

We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$,…

We study positive solutions of the pseudoparabolic equation with a sublinear source in $\mathbb{R}^n$. In this work, the source coefficient could be unbounded and time-dependent. Global existence of solutions to the Cauchy problem is…

We prove H\"older continuity of weak solutions of the uniformly elliptic and parabolic equations %$\Delta u-\frac{A}{|x|^{2+\beta}}u=0,\,\,(\beta\geq 0)$, and variable second order term coefficients case $%% \begin{equation}\label{01}…

The Cauchy problem in $\mathbb R^n$ is considered for \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v),\\ 0 = \Delta v + u. \end{array} \right. \end{eqnarray*} For each $n\ge 10$, a statement on stability…