This article is concerned with the asymptotic limits of the solutions of the homogeneous Dirichlet problem associated to a doubly nonlinear evolution equation of the form $u_{t} = \Delta_{p}u^{m} + g$, in a bounded domain, as the parameters $p$ and $m$ tend to infinity. We will address the limits in $p$ and $m$ separately and in sequence, eventually completing a convergence diagram for this problem. We prove, under additional assumptions on the domain and initial data, that the equation satisfied at the limit is independent of the order in which we take the limits in $p$ and $m$.

In this article, we develop the theory of weighted $L^2$ Sobolev spaces on unbounded domains in $\mathbb{R}^n$. As an application, we establish the elliptic theory for elliptic operators and prove trace and extension results analogous to the bounded, unweighted case.

Under general $p,q$-growth conditions, we prove that the Dirichlet problem \begin{equation*} \left\{ \begin{array}{ll} \displaystyle \sum_{i=1}^{n}\frac{\partial }{\partial x_{i}}a^{i}(x,Du)=b(x) & \quad \text{in}\,\Omega , \\ u=u_{0} & \quad \text{on}\,\partial \Omega \end{array} \right. \end{equation*} has a weak solution $u\in W_{\mathrm{loc}}^{1,q} \Big ( \Omega \Big ) $ under the assumptions $ 1 < p\leq q\leq p+1$ and $q < p\tfrac{n-1}{n-p}. $ More regularity applies. Precisely, this solution is also in the class $W_{ \text{loc}}^{1,\infty }(\Omega )\cap {W_{\text{loc}}^{2,2}(\Omega )}$.

We consider space-periodic evolutionary and travelling-wave solutions to the regularized long-wave equation (RLWE) with damping and forcing. We establish existence, uniqueness and smoothness of the evolutionary solutions for smooth initial conditions, and global in time spatial analyticity of such solutions for analytical initial conditions. The width of the analyticity strip decays at most polynomially. We prove existence of travelling-wave solutions and uniqueness of travelling waves of a sufficiently small norm. The importance of damping is demonstrated by showing that the problem of finding travelling-wave solutions to the undamped RLWE is not well-posed. Finally, we demonstrate the asymptotic convergence of the power series expansion of travelling waves for a weak forcing.

In this paper, we consider the problem $$ \left\{ \begin{array}{ll} -\Delta u=|x|^{\alpha}u^p & \text{ in } B ,\\ u>0 & \hbox{ in } B ,\\ u=0 & \hbox{ on }\partial B , \end{array}\right. $$ where $ B $ is the unit ball of $\mathbb R^N$, $N\ge 3$, $p > 1$ and $0 < \alpha\leq 1$. We prove the existence of (at least) one branch of non-radial solutions that bifurcate from the radial ones and that this branch is unbounded.

The aim of this work is to extend a result by Suzuki and Watson concerning an inverse property for caloric functions. Our result applies, in particular, to the heat operator on stratified Lie groups and to Kolmogorov-Fokker-Planck-type operators. We show that the open sets characterizing the solutions to the involved equations, in terms of suitable average operators, have to be the level sets of the fundamental solutions of the relevant operators. The technique adopted exploits the structure of the propagation sets, i.e., the sets where the solutions to the involved equations attain their maximum.