For the nonlinear heat equation with a fractional Laplacian $u_t+{(-\Delta )}^{\alpha /2}u=u^2$, 1 < α ≤ 2, the first initial-boundary value problem in a disk is considered. Small initial conditions, homogeneous boundary conditions, and periodicity conditions in the angular coordinate are imposed. Existence and uniqueness of a global-in-time solution is proved, and the solution is constructed in the form of a series of eigenfunctions of the Laplace operator in the disk. First-order long-time asymptotics of the solution is obtained....

The nonlinear eigenvalue problem for p-Laplacian $$\left\{\begin{array}{c}-{div\left(a\left(x\right)\right|\nabla u|}^{p-2}{\nabla u)=\lambda g\left(x\right)|u|}^{p-2}u\text{in}{ℝ}^N,u>0\text{in}{ℝ}^N,\underset{\left|x\right|\to \infty }{lim}u\left(x\right)=0,\hfill \end{array}\right.$$ is considered. We assume that $1<p<N$ and that $g$ is indefinite weight function. The existence and $C^{1,\alpha }$-regularity of the weak solution is proved.

The nonlinear dissipative wave equation $u_{tt}-\Delta u+{\left|u_t\right|}^{h-1}{u}_t=0$ in dimension $d\&gt;1$ has strong solutions with the following structure. In $0\le t\&lt;1$ the solutions have a focusing wave of singularity on the incoming light cone $\left|x\right|=1-t$. In $\{t\ge 1\}$ that is after the focusing time, they are smoother than they were in $\{0\le t\&lt;1\}$. The examples are radial and piecewise smooth in $\{0\le t\&lt;1\}$

In this paper, we present a nonlinear model for laser-plasma interaction describing the Raman amplification. This system is a quasilinear coupling of several Zakharov systems. We handle the Cauchy problem and we give some well-posedness and ill-posedness result for some subsystems.

In this paper we present several nonlinear models of suspension bridges; most of them have been introduced by Lazer and McKenna. We discuss some results which were obtained by the authors and other mathematicians for the boundary value problems and initial boundary value problems. Our intention is to point out the character of these results and to show which mathematical methods were used to prove them instead of giving precise proofs and statements.

We consider a class of nonlinear parabolic problems where the coefficients are depending on a weighted integral of the solution. We address the issues of existence, uniqueness, stationary solutions and in some cases asymptotic behaviour.