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In this paper, we consider the following nonlocal parabolic equation \begin{equation*} u_{t}-\Delta u=\left( \int_{\Omega}\frac{|u(y,t)|^{2^{\ast}_{\mu}}}{|x-y|^{\mu}}dy\right) |u|^{2^{\ast}_{\mu}-2}u,\ \text{in}\ \Omega\times(0,\infty),…

We consider the stochastic heat equation $\partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u$, with a smooth space-time stationary Gaussian random field $V(s,y)$, in dimensions $d\geq 3$, with an initial condition…

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…

We prove that the Cauchy problem associated with the one dimensional quadratic (fractional) heat equation: $u_t=D_x^{2\alpha} u \mp u^2,\; t\in (0,T),\; x\in \R$ or $ \T $, with $ 0<\alpha\le 1 $ is well-posed in $ H^s $ for $ s\ge…

We consider nonlinear parabolic SPDEs of the form $\partial_t u=-(-\Delta)^{\alpha/2} u + b(u) +\sigma(u)\dot w$, where$\dot w$ denotes space-time white noise. The functions $b$ and $\sigma$ are both locally Lipschitz continuous. Under some…

We are concerned with nonnegative solutions to the Cauchy problem for the porous medium equation with a variable density $\rho(x)$ and a power-like reaction term $u^p$ with $p>1$. The density decays {\it fast} at infinity, in the sense that…

This paper estimates the blow-up time for the heat equation $u_t=\Delta u$ with a local nonlinear Neumann boundary condition: The normal derivative $\partial u/\partial n=u^{q}$ on $\Gamma_{1}$, one piece of the boundary, while on the rest…

We study the existence of sign-changing solutions to the nonlinear heat equation $\partial _t u = \Delta u + |u|^\alpha u$ on ${\mathbb R}^N $, $N\ge 3$, with $\frac {2} {N-2} < \alpha <\alpha _0$, where $\alpha _0=\frac {4} {N-4+2\sqrt{…

This paper is concerned with a Cauchy problem for the three-dimensional (3D) nonhomogeneous incompressible heat conducting magnetohydrodynamic (MHD) equations in the whole space. First of all, we establish a weak Serrin-type blowup…

We consider the Cauchy problem for the focusing Hartree equation $iu_{t}+\Delta u+(|\cdot|^{-3}\ast|u|^{2})u=0$ in $\mathbb{R}^{5}$ with the initial data in $H^1$, and study the divergent property of infinite-variance and nonradial…

In this paper we will see that the global or local existence of solutions to \begin{eqnarray*} \dfrac{\partial u_{1}}{\partial t} & = & \mathit{k}_{1} (t) \Delta u_{1} + h_{1}(t) u_{1}^{p_{11}} u_{2}^{p_{12}},\\ \dfrac{\partial…

We show that any finite energy solution of the energy-critical nonlinear heat flow in dimensions $d\geq 3$ asymptotically resolves into a sum of possibly time-dependent solitons, a radiation term, and an error term that vanishes in the…

In this paper, we study the Cauchy problem for a heat equation governed by a mixed local--nonlocal diffusion operator with spatially irregular coefficients. We first establish classical well-posedness in an energy framework for bounded,…

It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations $\square u + u = F(u,{\partial u})$ with cubic nonlinearities. However, for the short pulse initial data $(u, \partial_tu)(0,…

We consider a family of nonlinear stochastic heat equations of the form $\partial_t u=\mathcal{L}u + \sigma(u)\dot{W}$, where $\dot{W}$ denotes space-time white noise, $\mathcal{L}$ the generator of a symmetric L\'evy process on $\R$, and…

We study the existence and nonexistence of a Cauchy problem of the semilinear heat equation $\partial_tu=\Delta u+|u|^{p-1}u$ in $\mathbb{R}^N\times(0,T)$, $u(x,0)=\phi(x)$ in $\mathbb{R}^N$, in $L^1(\mathbb{R}^N)$. Here, $N \ge 1$,…

We consider the energy critical four dimensional semi linear heat equation \partial tu-\Deltau-u3 = 0. We show the existence of type II finite time blow up solutions and give a sharp description of the corresponding singularity formation.…

In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation with time-dependent absorption $u_{t}-\Delta_{\mathbb{H}}u=- k(t)u^p$ posed on $\mathbb{H}^n$, driven by the Heisenberg Laplacian and supplemented…

We study the existence of nontrivial nonlocal nonnegative solutions $u(x,t)$ of the nonlinear initial value problems \[ (\partial_t -\Delta)^\alpha u\geq u^\lambda \quad \text{in } \mathbb{R}^n \times\mathbb{R},\,n\geq 1 \] \[ u=0…

In this paper, we deal with the Cauchy problem of the quasilinear Sch\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{lll} iu_t=\Delta u+2uh'(|u|^2)\Delta h(|u|^2)+(W(x)\ast|u|^2)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0)=u_0(x),\quad…