Related papers: An inhomogeneous nonlinear parabolic equation with…

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…

In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=\Delta u+v^p,\qquad & x\in\Omega,\,\,\,t>0,\\ \partial_t…

A general class of probability density functions \[u(x,t)=Ct^{-\alpha d}\left (1-\left (\frac{\|x\|}{ct^{\alpha}}\right )^{\beta}\right )_+^{\gamma},\quad x\in \mathbb{R}^d,t>0,\] is considered, containing as particular case the Barenblatt…

We study inhomogeneous semilinear parabolic equations with source term f independent of time u_{t}={\Delta}u+u^{p}+f(x) on a metric measure space, subject to the conditions that f(x)\geq 0 and u(0,x)=\phi(x)\geq 0. By establishing…

In this paper we deal with parabolic problems whose simplest model is $$ \begin{cases} u'- \Delta_{p} u + B\frac{|\nabla u|^p}{u} = 0 & \text{in} (0,T) \times \Omega,\newline u(0,x)= u_0 (x) &\text{in}\ \Omega, \newline u(t,x)=0 &\text{on}\…

We show a triviality result for "pointwise" monotone in time, bounded "eternal" solutions of the semilinear heat equation \begin{equation*} u_{t}=\Delta u + |u|^{p} \end{equation*} on complete Riemannian manifolds of dimension $n \geq 5$…

In this paper, we prove sharp gradient estimates for a positive solution to the heat equation $u_t=\Delta u+au\log u$ in complete noncompact Riemannian manifolds. As its application, we show that if $u$ is a positive solution of the…

This paper concerns the global in time existence of solutions for a semilinear heat equation \begin{equation} \tag{P} \label{eq:P} \begin{cases} \partial_t u = \Delta u + f(u), &x\in \mathbb{R}^N, \,\,\, t>0, \\[3pt] u(x,0) = u_0(x) \ge 0,…

We study the problems of uniqueness for Hardy-H\'enon parabolic equations, which are semilinear heat equations with the singular potential (Hardy type) or the increasing potential (H\'enon type) in the nonlinear term. To deal with the…

We investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type $e^{\mu t}u^p$ ($\mu\in\mathbb{R}, p>1$), posed on cones of the hyperbolic space. Under a…

In this paper, we prove existence of stationary nontrivial homogeneous solutions for SQG equation with infinite energy (unbounded at the infinity). Our analysis also covers the existence of stationary solutions for generalized De Gregorio…

We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation $u_t-\Delta u=|x|^\alpha |u|^{p}+\zeta(t)\,{\mathbf w}(x)$ in $(0,\infty)\times\mathbb{R}^N$, where $N\geq 3$, $p>1$, $\alpha>-2$, $\z, {\mathbf…

We provide global gradient estimates for solutions to a general type of nonlinear parabolic equations, possibly in a Riemannian geometry setting. Our result is new in comparison with the existing ones in the literature, in light of the…

In the first part of this paper, we prove the global regularity, in an adequate parabolic Bessel-Potential space and then in the corresponding parabolic fractional Sobolev space, of the unique solution to following fractional heat equation…

We consider the problem \[-\Delta u + W(x)u = ((1/{|x|^{\alpha}} * |u|^{p}) |u|^{p-2}u, u \in H_{0}^{1}(\Omega)\], where $\Omega$ is an exterior domain in $\mathbb{R}^{N}$, $N\geq3,$ $\alpha \in(0,N)$, $p\in[2,(2N-\alpha)/(N-2)$, $W$ is…

The purpose of this work is to analyze the well-posedness and blow-up behavior of solutions to the nonlocal semilinear parabolic equation with a forcing term: \[ \partial_t u - \Delta u = \|u(t)\|_{q}^\alpha |u|^p + t^{\varrho}…

We prove the existence of global solutions to the nonlinear wave equation in $\mathbb{R}^{1+3}$ $$\Phi_{tt} - \Delta \Phi \pm \Phi|\Phi|^{p-1} = 0$$ in the energy-supercritical regime $p>5$, for a class of large initial data. Our initial…

In this paper, we consider the following indefinite fully fractional heat equation involving the master operator \begin{equation} (\partial_t -\Delta)^{s} u(x,t) = x_1u^p(x,t)\ \ \mbox{in}\ \R^n\times\R , \end{equation} where $s\in(0,1)$,…

Analytic solutions for cylindrical thermal waves in solid medium is given based on the nonlinear hyperbolic system of heat flux relaxation and energy conservation equations. The Fourier-Cattaneo phenomenological law is generalized where the…

In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equation \[ (\Delta -q(x,t)-\partial_t)u(x,t)=A(u(x,t)),\quad (x,t)\in M\times [0,T]. \] We establish space-time gradient estimates for positive…