We prove that the constant anisotropic mean curvature immersion of the sphere $S^2$ in $ \mathbb R^3$ is unique, provided that anisotropy is weak in the sense that the energy density function is close to the isotropic one.

We study the asymptotic behaviour of solutions to the Cauchy problem for linear degenerate parabolic equations. The coefficients of the equations may grow or tend to zero as $|x|\rightarrow \infty$. We establish necessary and sufficient conditions on the initial data which guarantee the stabilization of the solution to a constant.

We consider the Cauchy problem of the heat equation with a radially symmetric, negative potential $-V$ which behaves like $V(r)=O(r^{-\kappa})$ as $r\to\infty$, for some $\kappa>2$, and study the relation between the large-time behavior of hot spots of the solutions and the behavior of the potential at the space infinity. In particular, we prove that the hot spots tend to the space infinity as $t\to\infty$ and how their rates depend on whether $V(|\cdot|)\in L^1({\bf R}^N)$ or not.

We study the asymptotic behavior, as $t\to\infty$, of the solutions to the evolutionary $p$-Laplace equation \[ v_t={\operatorname{div}}( | {\nabla v} | ^{p-2}\nabla v), \] with time-independent lateral boundary values. We obtain the sharp decay rate of $\max_{x\in{\Omega}} | {v(x,t)-u(x)}| $, where $u$ is the stationary solution, both in the degenerate case $p > 2$ and in the singular case $1 < p > 2$. A key tool in the proofs is the Moser iteration, which is applied to the difference $v(x,t)-u(x)$. In the singular case, we construct an example proving that the celebrated phenomenon of finite extinction time, valid for $v(x,t)$ when $u\equiv 0$, does not have a counterpart for $v(x,t)-u(x)$.

The paper is concerned with linear thermoelastic plate equations in a domain $\Omega$: $$ u_{tt} + \Delta^2u+ \Delta \theta = 0 \ \text{ and }\ \theta_t - \Delta\theta - \Delta u_t = 0 \quad\text{in $\Omega\times(0, \infty)$}, $$ subject to the Dirichlet boundary condition $u|_\Gamma = D_\nu u|_\Gamma = \theta|_\Gamma = 0$ and initial condition $(u, u_t, \theta)|_{t=0} = (u_0, v_0, \theta_0) \in W^2_{p, D}(\Omega)\times L_p\times L_p$. Here, $\Omega$ is a bounded or exterior domain in ${{\mathbb R}}^n$ ($n\geq2$). We assume that the boundary $\Gamma$ of $\Omega$ is a $C^4$ hypersurface and we define $W^2_{p, D}$ by the formula $W^2_{p, D} = \{ u \in W^2_p : u|_\Gamma = D_\nu u|_\Gamma = 0\}$. We show that, for any $p \in (1, \infty)$, the associated semigroup $\{T(t)\}_{t\geq0}$ is analytic. Moreover, if $\Omega$ is bounded, then $\{T(t)\}_{t\geq0}$ is exponentially stable.

We study the orbital stability of single-spike semiclassical standing waves of a nonhomogeneous in space nonlinear Schrödinger-Poisson equation. When the nonlinearity is subcritical or supercritical we prove that the nonlocal Poisson-term does not influence the stability of standing waves, whereas in the critical case it may create instability if its value at the concentration point of the spike is too large. The proofs are based on the study of the spectral properties of a linearized operator and on the analysis of a slope condition. Our main tools are perturbation methods and asymptotic expansion formulas.

We study the asymptotic behavior, as $t\to\infty$, of bounded solutions to a second-order integro-differential equation in finite dimensions where the damping term is of memory type and can be of arbitrary fractional order less than 1. We derive appropriate Lyapunov functions for this equation and prove that any global bounded solution converges to an equilibrium of a related equation, if the nonlinear potential ${\mathcal E}$ occurring in the equation satisfies the Łojasiewicz inequality.

Let ${\Omega}\subset{\mathbb{R}}^N$ be a bounded domain with a Lipschitz boundary and let ${\sigma}$ be the restriction to ${\partial \Omega}$ of the $(N-1)$-dimensional Hausdorff measure. Let $B:\;{\partial \Omega}\times{\mathbb{R}}\to [0,+\infty]$ be ${\sigma}$-measurable in the first variable and assume that for ${\sigma}$-almost every $x\in{\partial \Omega}$, $B(x,\cdot)$ is a proper, convex, lower semicontinuous functional. We prove that, for every $p\in [2, N)$, the nonlinear submarkovian semigroup $S_{B,p}$ on $L^2({\Omega})\times L^2({\partial \Omega})$ generated by the operator $\Delta_p:=\mbox{div}(|\nabla u|^{p-2}\nabla u)$ with the nonlinear Wentzell-Robin type boundary conditions \[\Delta_pu+|\nabla u|^{p-2}\frac{\partial u}{\partial n} +\beta(\cdot,u)\ni 0\;\;\mbox{ on }\;{\partial \Omega},\] satisfies the following H\"older-continuity type property: for every $q\ge 2$, and for all $U_0, V_0\in L^q({\Omega})\times L^q({\partial \Omega})$ the following holds: \[\|S_{B,p}(t)U_0-S_{B,p}(t)V_0\|_{L^\infty({\Omega})\times L^\infty({\partial \Omega})}\le C_1e^{C_2t}t^{-\alpha}\|U_0-V_0\|_{L^q({\Omega})\times L^q({\partial \Omega})} ^{\gamma},\] where $C_1$ and $C_2$ are positive constants depending only on $p, q, N$ and ${\Omega}$ and $\alpha, \gamma$ are suitable positive constants depending only on $p, q$ and $N$.