This paper is concerned with the orbital stability for solitary waves of a system of coupled KdV-equations. Although the coupled equations are not in Hamiltonian form, we can apply the abstract results of Grillakis et al. [4, 5] and use a detailed spectral analysis to obtain the stability of the solitary waves.

The existence and nonexistence of positive classical solutions is discussed for $-\Delta u = K(x) (1-|x|)^{- \alpha} u^{\beta}$ in the unit ball $B$ with Dirichlet boundary condition ${u |_{\partial B}= 0 }$. Our main tools are based on the variational method and Pohozaev's identity. The singularity of coefficients on the boundary will be handled with the symmetry of functions and some approximation procedures.

The differential equation $d(Mu(t))/dt=-Lu(t)+L_1u(t-1), $ $t\geq0, $ $u(t)=\varphi(t),$ $ -1\leq t\leq0$, for a given strongly continuous $X$-valued function $\varphi$ on $[-1,0]$ is studied, where $M, L, L_1$ are closed linear operators from the complex Banach space $X$ into itself, and $L$ is invertible. Though already in the finite dimensional case in general existence of continuous solutions on $[-1,\infty)$ may fail or it is possible to have continuous solutions only on a finite interval, we indicate classes of operators for which existence results analogous to the ones for regular equations $M=I$ hold. In particular, solutions are given explicitly by a recovery formula.

The nonlinear abstract differential equation $$ \frac{d}{dt}(Ay)+By(t)=F(t,Ky),\quad 0\le t\le\tau, $$ where $A,B,K$ are linear closed operators from a complex Banach space $Y$ into a Banach space $X$ is considered. The main assumption reads that the point $\xi =0$ is a polar singularity of the resolvent $(T-\xi I)^{-1}$, where $T=A(\lambda A+B)^{-1}$, $\lambda$ being a regular point of the operator pencil $\lambda A+B$. Mainly the case of a simple pole and of a second order pole are considered. Some examples of application to concrete partial differential equations are given. In particular, we show that the results work for mathematical models of nonlinear electrical networks.

We study heat kernel bounds, regularity on space variables and the holomorphic functional calculus on $L^p$ for operators of type $bA$ where $b$ is a complex bounded function and $A$ is a second-order elliptic operator.

This paper is concerned with a nonlinear integro-partial differential equation perturbed by a state dependent white noise arising from the aeroelastic panel vibration. With the aid of a stochastic energy equation, the existence, uniqueness and regularity of solutions are proved.

Integral and supremum estimates are proven for local solutions of degenerate and singular quasilinear parabolic equations. This is done with the aid of a local energy inequality and estimates in $L^{\text{weak}}_q$ spaces.