Related papers: Complex valued semi-linear heat equations in super…

By a probabilistic method we provide an explicit fundamental solution of the Cauchy problem associated to the heat equation on the half-line with constant drift and Dirichlet boundary condition at zero.

We consider the Cauchy problem for the Hamilton-Jacobi equation with critical dissipation, $$ \partial_t u + (-\Delta)^{ 1/2} u = |\nabla u|^p, \quad x \in \mathbb R^N, t > 0, \qquad u(x,0) = u_0(x) , \quad x \in \mathbb R^N, $$ where $p >…

The Cauchy problem for the two dimensional compressible Euler equations with data in the Sobolev space $H^s(\mathbb R^2)$ is known to have a unique solution of the same Sobolev class for a short time, and the data-to-solution map is…

We consider the second order Cauchy problem $$u''+m(|A^{1/2}u|^2)Au=0, u(0)=u_{0}, u'(0)=u_{1},$$ where $m:[0,+\infty)\to[0,+\infty)$ is a continuous function, and $A$ is a self-adjoint nonnegative operator with dense domain on a Hilbert…

In this paper, we apply a self-similar transformation to convert the parabolic equation with a Sobolev-Hardy term \begin{align*} u_t-\Delta u= \frac{|u|^{q-2}u}{\left|x\right|^s} & \text { in } \mathbb{R}^N \times(0, \infty), \end{align*}…

We study the dynamics of the following porous medium equation with strong absorption $$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$ posed for $(t, x) \in (0,\infty) \times \mathbb{R}^N$, with $m > 1$, $q \in (0, 1)$ and $\sigma >…

In this paper we show existence of finite energy solutions for the Cauchy problem associated with a semilinear wave equation with interior damping and supercritical source terms. The main contribution consists in dealing with…

In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped $\sigma$-evolution models with different power nonlinearities. By assuming additional $L^m$ regularity on the initial data, with $m…

We consider the semilinear heat equation $$\partial_t u -\Delta u =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in \mathbb{R}$. We first show an…

We consider the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation \[iu_{t} +\Delta u=|x|^{-b} f\left(u\right), u\left(0\right)=u_{0} \in H^{s} (\mathbb R^{n}),\] where $0<s<\min \left\{n,\;\frac{n}{2}…

In this paper, we consider blow-up of solutions to the Cauchy problem for the following fractional NLS, $$ \textnormal{i} \, \partial_t u=(-\Delta)^s u-|u|^{2 \sigma} u \quad \text{in} \,\, \R \times \R^N, $$ where $N \geq 2$, $1/2 <s<1$…

Let $\mathbb G$ be a graded Lie group with homogeneous dimension $Q$. In this paper, we study the Cauchy problem for a semilinear hypoelliptic damped wave equation involving a positive Rockland operator $\mathcal{R}$ of homogeneous degree…

Let $G=(V,E)$ be a finite or locally finite connected weighted graph, $\Delta$ be the usual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence of global solutions for the following semilinear heat equation…

We study the homogeneous equation (*) $ u' = \Delta u$, $t > 0$, $u(0)=f\in wX$, where $wX$ is a weighted Banach space, $w(x)= (1+||x||)^k$, $x\in \r^n$ with $k\ge 0$, $ \Delta$ is the Laplacian, $Y$ a complex Banach space and $X$ one of…

We construct a singular solution of a stationary nonlinear Schr\"{o}dinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity…

We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted H\"older and Sobolev spaces with discrete…

Let $\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $\Delta$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $\lambda_{0}$, and $\sigma \in (0,1)$. The aim of this paper is twofold. On the…

In this paper, we are concerned with the Cauchy problem for the reaction-diffusion equation $\partial_t u+t^\beta\mathcal{L} u= - h(t)u^p$ posed on $\mathbb{R}^N$, driven by the mixed local-nonlocal operator…

In this paper we study the Cauchy problem for semi-linear de Sitter models with power non-linearity. The model of interest is \[ \phi_{tt} - e^{-2t} \Delta \phi + n\phi_t+m^2\phi=|\phi|^p,\quad (\phi(0,x),\phi_t(0,x))=(f(x),g(x)),\] where…

We consider a non-linear heat equation $\partial_t u = \Delta u + B(u,Du)+P(u)$ posed on the $d$-dimensional torus, where $P$ is a polynomial of degree at most $3$ and $B$ is a bilinear map that is not a total derivative. We show that, if…