Related papers: Feller evolution families and parabolic equations …

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$,…

In this paper, we study some regularity issues concerning the gradient of weak solutions of $u_t - {\rm div} \mathcal{A}(x,t,\nabla u) = g$, where $\mathcal{A}(x,t,\nabla u)$ is modeled after the $p$-Laplace operator. The main results we…

In this paper we study the existence and summability of the solutions to the following parabolic-elliptic system of partial differential equations with discontinuous coefficients: \begin{equation*} \begin{cases} u_t -…

The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem $$ \partial_t u-\Delta u=a u-b(x) u^p \text{in} \Omega\times \R^+, u(0)=u_0, u(t)|_{\partial \Omega}=0 $$ as $p\to +\infty$, where…

We obtain sufficient conditions for the existence and uniqueness of solutions with non-negative components to general quasilinear parabolic problems \begin{equation*} \partial_t u^k = \sum_{i,j=1}^n a_{ij} (t,x,u)\partial^2_{x_i x_j}\!u^k +…

We give a substitute to Feller property for semigroups of time-changed processes; under some conditions this leads to establish sufficient (new) conditions for the semigroups to be Feller. Moreover, given a standard process and a sequence…

Let $p$ and $q$ be locally H\"{o}lder functions in $\RR^N$, $p>0$ and $q\geq 0$. We study the Emden-Fowler equation $-\Delta u+ q(x)|\nabla u|^a=p(x)u^{-\gamma}$ in $\RR^N$, where $a$ and $\gamma$ are positive numbers. Our main result…

We study the behavior of weak solutions to the singular quasilinear elliptic problem $-\Delta_p u + \vartheta |\nabla u|^q = \frac{1}{u^\gamma} + f(u)$, in a bounded domain with the Dirichlet boundary condition, where $p>1$, $\gamma>0$,…

We face the well-posedness of linear transport Cauchy problems $$\begin{cases}\dfrac{\partial u}{\partial t} + b\cdot\nabla u + c\,u = f&(0,T)\times{\mathbb R}^n\\u(0,\cdot)=u_0\in L^\infty&{\mathbb R}^n\end{cases}$$ under borderline…

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations \begin{eqnarray*} \Delta^{2}u-M(\|\nabla u\|_{2}^{2})\Delta u+V(x)u=f(x,u),\ \ \ \ \ x\in \mathbb{R}^{N},…

In this paper, we consider the backward problem for fractional in time evolution equations $\partial_t^\alpha u(t)= A u(t)$ with the Caputo derivative of order $0<\alpha \le 1$, where $A$ is a self-adjoint and bounded above operator on a…

We discuss the occurrence of positive solutions which decay to 0 as $| x|\to+\infty$ to the differential equation $\Delta u+f(x,u)+g(| x|)x\cdot\nabla u=0$, $| x|>R>0$, $x\in\mathbb{R}^{n}$, where $n\geq 3$, $g$ is nonnegative valued and…

In this paper we present a Calder\'{o}n-Zygmund approach for a large class of parabolic equations with pseudo-differential operators $\mathcal{A}(t)$ of arbitrary order $\gamma\in(0,\infty)$. It is assumed that $\cA(t)$ is merely measurable…

We consider a class of nonlinear parabolic equations \[ \dfrac{\partial}{\partial t} b(u)-\nabla \cdot (A(x,t,u,\nabla u))+H(x,t,\nabla u)=f , \] where $H$ is a nonlinear lower order term satisfied the Carath$\acute{e}$odory condition and…

Suppose that $d\geq 1$ and $0<\beta<2$. We establish the existence and uniqueness of the fundamental solution $q^b(t, x, y)$ to the operator $\mathcal{L}^b=\Delta+S^b$, where $$S^bf(x) := \int_{\mathbb{R}^d} \left( f(x+z) - f(x) - \nabla…

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$,…

We consider a parabolic equation of the form u_t=\Delta u +f(u)+h(x,t) in R^N\times (0,\infty), where f in C^1(R) is such that f(0)=0 and f'(0)<0 and h is a suitable function on R^N\times (0,\infty). We show that under certain conditions,…

For the time-space fractional degenerate Keller-Segel equation \begin{equation*} \begin{cases} \partial _{t}^{\beta }u=-(-\Delta )^{\frac{\alpha}{2}}(\rho (v)u),& t>0\\ (-\Delta )^{\frac{\alpha}{2}} v+v=u,& t>0 \end{cases} \end{equation*}…

We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation $u_t-\Delta _{p(x)}u = f(x,u)$ in $ (0,T)\times\Omega$; $u = 0$ on $(0,T)\times\partial\Omega$; $u(0,x)=u_0(x)$ in $\Omega$;…

We prove an extension theorem for local solutions of the 3d incompressible Euler equations. More precisely, we show that if a smooth vector field satisfies the Euler equations in a spacetime region $\Omega\times(0,T)$, one can choose an…