The goal of this paper is to study qualitative properties of solutions to the Cauchy problem for structurally damped $\sigma-$evolution models \begin{align*} & u_{tt}+(-\Delta)^\sigma u +b(t)(-\Delta)^\delta u_{t}=0, \\ & u(0,x)=u_{0}(x),\,\,\,\, u_t(0,x)=u_{1}(x), \end{align*} where $\sigma>1$, $\delta \in(0,\sigma)$, and the dissipation coefficient $b=b(t)$ is a time-dependent and strictly increasing positive function. On the one hand, we are interested in $L^p-L^q$ estimates for the energies of higher order. On the other hand, we are interested in Gevrey smoothing properties of solutions. Finally, we prove the optimality of decay by using scale-invariant models. The main tool of our considerations is a related WKB-analysis.

We consider solutions of the Poisson equation on a family of domains with holes shrinking to a point. Assuming Robin or Neumann boundary conditions on the boundary of the holes, we show that the solution converges uniformly to the solution of the Poisson equation on the domain without the holes. This is in contrast to Dirichlet boundary conditions where there is no uniform convergence. The results substantially improve earlier results on $L^p$-convergence. They can be applied to semi-linear problems.

The work is devoted to Dirichlet problem for sub-quintic semi-linear wave equation with damping term of the form $(-{\Delta_x})^\alpha{\partial_t}u$, $\alpha\in(0,\frac{1}{2})$, in bounded smooth domains of $\mathbb R^3$. It appears that to prove well-posedness and develop smooth attractor theory for the problem, we need additional regularity of the solutions, which does not follow from the energy estimate. Considering the original problem as perturbation of the linear one the task is reduced to derivation of Strichartz type estimate for the linear wave equation with fractional damping, which is the main feature of the work. Existence of smooth exponential attractor for the natural dynamical system associated with the problem is also established.

In this article, we analyze solutions of the following system of elliptic differential equations \begin{equation*} \begin{cases} - \Delta u = \Lambda u + \nabla_u \eta(u, \Lambda) & \text{ in } \Omega\\ u = 0 & \text{ on } \partial \Omega. \end{cases} \end{equation*} We provide sufficient evidence to prove the existence of global bifurcation points of nontrivial solutions of this system. Moreover, we describe a symmetry breaking phenomenon that occurs on continua of nontrivial solutions of it.

In this article, the author studies the weak solutions for a sixth-order thin film equation in two space dimensions, which arises in the industrial application of the isolation oxidation of silicon. Based on the Schauder type estimates, we establish the global existence of classical solutions for regularized problems. Our approach lies in the combination of the energy techniques with some methods based on the framework of Campanato spaces. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions in two space dimensions. The nonnegativity of solutions is also discussed.

One of the simplest formulas of the calculus of several variables says that, when applied to spherical symmetric functions $u=u(\rho)$ in $\mathbb R^n$, the Laplace operator takes the form $\Delta u=u''(\rho)+(n-1)u'(\rho)/\rho$. In this paper, we derive the analogous explicit expression for the polyharmonic operator $\Delta^k$ in the case of spherical symmetry. Moreover, if $B$ is a ball centered at the origin and $u\in H^k_0(B)$ is spherical symmetric, then, we deduce the functional \[ J[u]= \begin{cases} \displaystyle \frac{1}{2}\int_{\Omega} (\Delta^{k/2} u(x))^2\,dx&\text{if $k$ is even}\cr \displaystyle \frac{1}{2}\int_{\Omega} |\nabla \Delta^{(k-1)/2} u(x)|^2\,dx&\text{if $k$ is odd},\cr \end{cases} \] of which $(-\Delta)^k$ is gradient as a sum of integrals in one variable. The results are based upon nontrivial combinatorial identities, which are proved by means of Zeilberger's algorithm.