In this paper we analyze movable singularities of the solutions of the equation for self-similar profiles resulting from semilinear wave equation. We study local analytic solutions around two fixed singularity points of this equation- ρ = 0 and ρ = 1. The movable singularities of local analytic solutions at the origin will be connected with those of the Lane-Emden equation. The function describing approximately their position on the complex plane will be derived. For ρ > 1 some topological considerations...

Sufficient conditions for the problem $$\frac{{\partial }^{2}u}{\partial x\partial y}=P_0(x,y)u+P_1(x,y)\frac{\partial u}{\partial x}+P_2(x,y)\frac{\partial u}{\partial y}+q(x,y),u(x+{\omega }_1,y)=u(x,y),u(x,y+{\omega }_2)=u(x,y)$$ to have the Fredholm property and to be uniquely solvable are established, where ${\omega }_1$ and ${\omega }_2$ are positive constants and $P_j:R^2\to R^{n\times n}$$(j=0,1...$

In this paper we consider the existence and asymptotic behavior of solutions of the following problem: $$u_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta u(t,x)+\delta |u_t{(t,x)|}^{p-1}{u}_t(t,x)={\mu \left|u(t,x)\right|}^{q-1}u(t,x),x\in \Omega ,t\ge 0,v_{tt}{(t,x)-(\alpha +\beta \parallel \nabla u(t,x)\parallel }_2^2+{\beta \parallel \nabla v(t,x)\parallel }_2^2)\Delta v(t,x)+\delta |v_t{(t,x)|}^{p-1}{v}_t(t,x)={\mu \left|v(t,x)\right|}^{q-1}v(t,x),x\in \Omega ,t\ge 0,u(0,x)=u_0\left(x\right),u_t(0,x)=u_1\left(x\right),x\in \Omega ,v(0,x)=v_0\left(x\right),v_t(0,x)=v_1\left(x\right),x\in \Omega ,u{{|}_{{}_{\partial \Omega }}=v|}_{{}_{\partial \Omega }}=0$$ where $q>1$, $p\ge 1$, $\delta >0$, $\alpha >0$, $\beta \ge 0$, $\mu \in ℝ$ and $\Delta$ is the Laplacian in ${ℝ}^N$.