We consider an initial boundary value problem related to the equation $u_t-\Delta u+{\int }_0^{t}g\left(t-s\right)\Delta u\left(x,s\right)ds={\left|u\right|}^{p-2}u$ and prove, under suitable conditions on $g$ and $p$, a blow-up result for certain solutions with positive initial energy.

We study the existence of nontrivial solutions for the problem $\Delta u=u$, in a bounded smooth domain $\Omega \subset {ℝ}^{\mathbb{N}}$, with a semilinear boundary condition given by $\partial u/\partial \nu =\lambda u-W\left(x\right)g\left(u\right)$, on the boundary of the domain, where $W$ is a potential changing sign, $g$ has a superlinear growth condition, and the parameter $\lambda \in \left]0,{\lambda }_1\right];{\lambda }_1$ is the first eigenvalue of the Steklov problem. The proofs are based on the variational and min-max methods.

We have investigated a dynamic thermoviscoelastic system (2003), establishing existence and uniqueness results for a related initial and boundary values problem. The aim of the present paper is to study the asymptotic behavior of the solution to the above problem as the power of the acceleration forces goes to zero. In particular, well-posedness and regularity results for the limit (quasistatic) problem are recovered.

Let $X$ be an infinite-dimensional real reflexive Banach space with dual space $X^{\ast }$ and $G\subset X$ open and bounded. Assume that $X$ and $X^{\ast }$ are locally uniformly convex. Let $T:X\supset D\left(T\right)\to 2^{X^{\ast }}$ be maximal monotone and $C:X\supset D\left(C\right)\to X^{\ast }$ quasibounded and of type $\left({\tilde{S}}_{+}\right)$. Assume that $L\subset D\left(C\right)$, where $L$ is a dense subspace of $X$, and $0\in T\left(0\right)$. A new topological degree theory is introduced for the sum $T+C$. Browder's degree theory has thus been extended to densely defined perturbations of maximal monotone operators while results of Browder and Hess have been extended to various classes of single-valued densely defined generalized pseudomonotone perturbations $C$. Although the main results are of theoretical nature, possible applications of the new degree theory are given for several other theoretical problems in nonlinear functional analysis.

We prove the existence of an obstacle lying on the bottom of an infinite channel inducing a surface on the upper bound of the fluid domain. This problem is the inverse of the free-surface problem flow which has been studied by several authors. We use the implicit function theorem to establish the existence of the solution of the problem.

We give one example for a one-parameter nonexpansive semigroup. This example shows that there exists a one-parameter nonexpansive semigroup $\left\{T\left(t\right):t\ge 0\right\}$ on a closed convex subset $C$ of a Banach space $E$ such that ${\text{lim}}_{t\to \infty }\Vert \left(1/t\right){\int }_0^{t}T\left(s\right)xds-x\Vert =0$ for some $x\in C$, which is not a common fixed point of $\left\{T\left(t\right):t\ge 0\right\}$.

We are concerned with the multiplicity of solutions of the following singularly perturbed semilinear elliptic equations in bounded domains $\Omega :-{\varepsilon }^2\Delta u+a(\cdot )u=u{\left|u\right|}^{p-2}$ in $\Omega$, $u>0$ in $\Omega$, $u=0$ on $\partial \Omega$. The main purpose of this paper is to discuss the relationship between the multiplicity of solutions and the profile of $a(\cdot )$ from the variational point of view. It is shown that if $a$ has a “peak” in $\Omega$, then (P) has at least three solutions for sufficiently small $\varepsilon$.