相关论文: Dynamics of time-periodic reaction-diffusion equat…
In this paper, we study the Cauchy problem to the linear damped $\sigma$-evolution equation with time-dependent damping in the effective cases \begin{equation*} u_{t t}+(-\Delta)^\sigma u+b(t)(-\Delta)^\delta u_t=0, \end{equation*} and…
The paper concerns front propagation for the following mono-stable reaction-diffusion-advection equation \[f(u)u_x + g(u)u_\tau = [d(u)|u_x|^{p-2} u_x]_x+ \rho(u), \quad (x,\tau)\in \R\times [0,+\infty).\] Besides existence and…
In this paper we investigate the following fractional order in time Cauchy problem \begin{equation*} \begin{cases} \mathbb{D}_{t}^{\alpha }u(t)+Au(t)=f(u(t)), & 1<\alpha <2, u(0)=u_{0},\,\,\,u^{\prime }(0)=u_{1}. & \end{cases}%…
It is shown that semilinear parabolic evolution equations $u'=A+f(t,u)$ featuring H\"older continuous nonlinearities $ f=f(t,u)$ with at most linear growth possess global strong solutions for a general class of initial data. The abstract…
This paper establishes the emergence of slowly moving transition layer solutions for the $p$-Laplacian (nonlinear) evolution equation, \[ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0, \] where $\varepsilon>0$…
We investigate a reaction-diffusion-advection equation of the form $u_t-u_{xx}+\beta u_x=f(u)$ $(t>0,\,0<x<h(t))$ with mixed boundary condition at $x=0$ and a free boundary condition at $x=h(t)$. Such a model may be applied to describe the…
The topic of this paper is a semi-linear, defocusing wave equation $u_{t t}-\Delta u=-|u|^{p-1} u$ in sub-conformal case in the higher dimensional space whose initial data are radical and come with a finite energy. We prove some decay…
This paper is devoted to the study of some qualitative and quantitative aspects of nonlinear propagation phenomena in diffusive media. More precisely, we consider the case a reaction-diffusion equation in a periodic medium with…
An operatorial based approach is used here to prove the existence and uniqueness of a strong solution $u$ to the time-varying nonlinear Fokker--Planck equation $u_t(t,x)-\Delta(a(t,x,u(t,x))u(t,x))+{\rm div}(b(t,x,u(t,x))u(t,x))=0$ in…
We study the asymptotic behaviour of solutions to the delayed monostable equation $(*)$: $u_{t}(t,x) = u_{xx}(t,x) - u(t,x) + g(u(t-h,x)),$ $x \in R,\ t >0,$ with monotone reaction term $g: R_+ \to R_+$. Our basic assumption is that this…
It is known that for some time periodic potentials $q(t, x) \geq 0$ having compact support with respect to $x$ some solutions of the Cauchy problem for the wave equation $\partial_t^2 u - \Delta_x u + q(t,x)u = 0$ have exponentially…
We consider the Cauchy problem \begin{align*} \partial_t u+u\partial_x u+L(\partial_x u) &=0, \\ u(0,x)=u_0(x) \end{align*} on the torus and on the real line for a class of Fourier multiplier operators $L$, and prove that the solution map…
We investigate the well-posedness of the fast diffusion equation (FDE) in a wide class of noncompact Riemannian manifolds. Existence and uniqueness of solutions for globally integrable initial data was established in [5]. However, in the…
We study entire solutions to homogeneous reaction-diffusion equations in several dimensions with Fisher-KPP reactions. Any entire solution $0<u<1$ is known to satisfy \[ \lim_{t\to -\infty} \sup_{|x|\le c|t|} u(t,x) = 0 \qquad \text{for…
In this article, we propose a novel front tracking scheme for scalar conservation laws with spatially heterogeneous, uniformly convex flux and prove that approximations converge to the unique entropy solution. The main tools are Dafermos'…
We consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_\xi f(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} with $q \in (0, \infty)$ in a…
In this paper, we consider the Cauchy problem {align*} \{{array}{ll}&i u_t+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=\phi(x)\in \Sigma, \quad x\in\mathbb{R}^N, {array}. {align*}…
This work aims to study the initial-boundary value problem of the reaction-diffusion equation with state-dependent delay $\pa_{t}u-\Delta u=f(u)+g(u,u(t-\tau(t,u_t)))+h(t,x)$ in a bounded domain. We establish the global existence of the…
We consider the Cauchy-problem for a parabolic equation of the following type: \begin{equation*} \frac{\partial u}{\partial t}= \Delta u+ f(u,|x|), \end{equation*} where $f=f(u,|x|)$ is supercritical. We supply this equation by the initial…
This work aims to study the initial-boundary value problem of the reaction-diffusion equation $\pa_{t}u-\Delta u=f(u)+g(u(t-\tau(t,u_t)))+h(t,x)$ in a bounded domain with state-dependent delay and supercritical nonlinearities. We establish…