Related papers: Critical non-linearity for some evolution equation…
In this paper we consider the critical exponent problem for the semilinear damped wave equation with time-dependent coefficients. We treat the scale invariant cases. In this case the asymptotic behavior of the solution is very delicate and…
In this paper, we would like to study the linear Cauchy problems for semi-linear $\sigma$-evolution models with mixing a parabolic like damping term corresponding to $\sigma_1 \in [0,\sigma/2)$ and a $\sigma$-evolution like damping…
We consider the Cauchy problem for the nonlinear Schr\"{o}dinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\R ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first…
We consider a nonlinear integro-differential equation for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The equation can be written as a quasilinear Cauchy problem. For bounded reaction…
We study the existence of solutions of a nonlinear parabolic problem of Cauchy-Dirichlet type having a lower order term which depends on the gradient. The model we have in mind is the following: \[ \begin{cases}\begin{split} &…
We study the nonlinear and nonlocal Cauchy problem \[ \partial_{t}u+\mathcal{L}\varphi(u)=0 \quad\text{in }\mathbb{R}^{N}\times\mathbb{R}_+,\qquad u(\cdot,0)=u_0, \] where $\mathcal{L}$ is a L\'evy-type nonlocal operator with a kernel…
We study the Cauchy problem for the fractional Schr\"{o}dinger equation $$ i\partial_tu = (m^2-\Delta)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < \alpha < 2$, and $F$ stands for the nonlinearity of…
This work is devoted to the study of a nonlocal-in-time evolutional problem for the first order differential equation in Banach space. Our primary approach, although stems from the convenient technique based on the reduction of a nonlocal…
We consider nonlinear parabolic equations involving fractional diffusion of the form $\partial_t u + (-\Delta)^s \Phi(u)= 0,$ with $0<s<1$, and solve an open problem concerning the existence of solutions for very singular nonlinearities…
The Aubin-Lions lemma and its variants play crucial roles for the existence of weak solutions of nonlinear evolutionary PDEs. In this paper, we aim to develop some compactness criteria that are analogies of the Aubin--Lions lemma for the…
We consider the Cauchy problem for the damped wave equations with variable coefficients a(x) having power type nonlinearity |u|^p. We discuss the global existence of solutions for small initial data and investigate the relation between the…
We study the Cauchy problem for systems of cubic nonlinear Klein-Gordon equations with different masses in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and…
In this paper, under the exponential/polynomial decay condition in Fourier space, we prove that the nonlinear solution to the quasi-periodic Cauchy problem for the weakly nonlinear Schr\"odinger equation in higher dimensions will…
In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form $$\begin{cases} &u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\…
We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The essence of the paper is that the constitutive relation, involving the…
We investigate some regularity properties of a class of doubly nonlinear anisotropic evolution equations whose model case is \begin{align*} \partial_t \big(|u|^{\alpha -1}u \big) - \sum^N_{i=1} \partial_i \big( |\partial_i u|^{p_i - 2}…
We study the Cauchy problem for a system of cubic nonlinear Klein-Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the order…
We discuss how the higher-order term $|u|^q$ $(q>1+2/(n-1))$ has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations $$ \partial_t^2 u-\Delta u=|v|^p, \qquad \partial_t^2…
Consider the equation u_t=\Delta u-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=\phi(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\frac\omega{|x|^2}$ as $|x|\to\infty$, for some $\omega\neq0$, and $a(x)$ is on the…
For the non-local space-time reaction-diffusion equation involving fractional $p$-Laplacian \begin{equation*} \begin{cases} \frac{\partial^{\alpha }u}{\partial t^{\alpha }}+(-\Delta)_{p}^{s} u=\mu u^{2}(1-kJ*u)-\gamma…