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We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional…

This paper is concerned with supersolutions to parabolic equations of the form \begin{equation} \partial_t U (x,t)-D(x)\Delta U(x,t)=0, \quad (x,t)\in \mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is positive.…

We develop a theory of existence and uniqueness for the following porous medium equation with fractional diffusion, $$ \{ll} \dfrac{\partial u}{\partial t} + (-\Delta)^{\sigma/2} (|u|^{m-1}u)=0, & \qquad x\in\mathbb{R}^N,\; t>0, [8pt]…

In this paper existence and nonexistence results of positive radial solutions of a Dirichlet $m$-Laplacian problem with different weights and a diffusion term inside the divergence of the form $\big(a(|x|)+g(u)\big)^{-\gamma}$, with…

The existence of nonnegative radially symmetric eternal solutions of exponential self-similar type $u(t,x)=e^{-p\beta t/(2-p)} f_\beta(|x|e^{-\beta t};\beta)$ is investigated for the singular diffusion equation with critical gradient…

In this paper, we study several inverse problems associated with a fractional differential equation of the following form: \[ (-\Delta)^s u(x)+\sum_{k=0}^N a^{(k)}(x) [u(x)]^k=0,\ \ 0<s<1,\ N\in\mathbb{N}\cup\{0\}\cup\{\infty\}, \] which is…

In this paper we study the existence of global-in-time energy solutions to the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity: $$u_{tt}-u_{xx} + \frac\mu{t}\,u_t = |u|^p \,, \quad t>t_0, \…

We consider the cauchy problem for the fractional power dissipative equation $u_t+(-\Delta )^{\beta/2} u=F(u)$, where $\beta>0$ and $F(u)=B(u, ...,u)$ and $B$ is a multilinear form on a Banach space $E$. We show a global existence result…

We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is…

Consider the quasilinear diffusion problem \[\begin{cases}\mathbf{u}'+\Pi(t,x,\mathbf{u},\Sigma \mathbf{u})\mathbb{A}\mathbf{u}=\mathbf{f}(t,x,\mathbf{u},\Sigma \mathbf{u})&\text{ in }]0,T[\times\Omega,\\\mathbf{u}=\mathbf{0}&\text{ in…

We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-\Delta)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first…

We study the nonlinear fractional reaction diffusion equation $\partial_{t}u + (-\Delta)^{s} u= f(t,x,u)$, $s\in(0,1)$ in a bounded domain $\Omega$ together with Dirichlet boundary conditions on $\R^N \setminus \Omega$. We prove asymptotic…

We construct a theory of existence, uniqueness and regularity of solutions for the fractional heat equation $\partial_t u +(-\Delta)^s u=0$, $0<s<1$, posed in the whole space $\mathbb{R}^N$ with data in a class of locally bounded Radon…

We investigate the nonlinear Schr\"{o}dinger equation $iu_{t}+\Delta u+|u|^{p-1}u=0$ with $1+\frac{4}{N}<p<1+\frac{4}{N-2}$ (when $N=1, 2$, $1+\frac{4}{N}<p<\infty$) in energy space $H^1$ and study the divergent property of…

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^\mu [\varphi (u)]=0$. Here $\mathcal{L}^\mu$ can be…

We consider the strategy of realizing the solution of a Cauchy problem with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge…

For $n\ge 3$, $0<m<\frac{n-2}{n}$, $\beta<0$ and $\alpha=\frac{2\beta}{1-m}$, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in…

We construct nonnegative weak solutions to the singular parabolic free boundary problem \[ \partial_t u - \Delta u = - \frac{\mathrm{d}}{\mathrm{d} u} u_+^\gamma , \] where $\gamma \in (0,1]$, $u_+ := \max\{u,0\}$, and the term in the…

In this paper we consider the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u + u_t = h(u);\qquad u(0;x) = f(x); \quad u_t(0;x) = g(x);$ where $h(s) = |s|^{1+2/n}\mu(|s|)$. Here n is the space dimension and $\mu$ is…

In this paper, by using variational methods and critical point theory, we shall mainly study the existence of infinitely many solutions for the following fractional Schr\"odinger-Maxwell equations $$( -\Delta )^{\alpha} u+V(x)u+\phi…