We consider the Cauchy problem in $\mathbb R^n$ for linear thermoelastic plate equations where heat conduction is modeled by either the Cattaneo law or by the Fourier law -- described by the relaxation parameter $\tau$, where $\tau>0$ corresponds to Cattaneo's law and $ \tau=0 $ corresponds to Fourier's law. Additionally, we take into account possible inertial effects characterized by a parameter $\mu\geq 0$, where $\mu=0$ corresponds to the situation without inertial terms. For the Catteneo system without inertial term, being a coupling of a Schrödinger type equation (the elastic plate equation) with a hyperbolic system for the temperature and the heat flux, we shall show that a regularity-loss phenomenon appears in the asymptotic behavior as time tends to infinity, while this is not given in the standard model where the Cattaneo law is replaced by the standard Fourier law. This kind of effect of changing the qualitative behavior when moving from Fourier to Cattaneo reflects the effect known for bounded domains, where the system with Fourier law is exponentially stable while this property is lost when going to the Cattaneo law. In particular, we shall describe in detail the singular limit as $\tau\to 0$. For the system with inertial term we demonstrate that it is of standard type, not of regularity loss type. The corresponding limit of a vanishing inertial term is also described. All constants appearing in the main results are given explicitly, allowing for quantitative estimates. The optimality of the estimates is also proved.

We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by regularized problems, and we also show the uniqueness and existence of a level set flow for bounded crystals.

In this paper, we investigate boundary blow-up solutions of the problem \begin{equation*} \quad \left\{ \begin{array}{l} -\Delta _{p(x)}u+f(x,u)=\rho (x,u)+K(x)|\nabla u|^{m(x)}\mbox{ in }\Omega , \\[2mm] u(x)\rightarrow +\infty \mbox{ as }d(x,\partial \Omega )\rightarrow 0, \end{array} \right. \end{equation*} where $\Delta _{p(x)}u=\mathrm{div}\,(|\nabla u|^{p(x)-2}\nabla u)$ is called $p(x)$-Laplacian. Our results extend the previous work of J. García-Melián, A. Suárez [23] from the case where $p(\cdot )\equiv 2 $, without gradient term, to the case where $p(\cdot )$ is a function, with gradient term. It also extends the previous work of Y. Liang, Q.H. Zhang and C.S. Zhao [38] from the radial case in the problem to the non-radial case. The existence of boundary blow-up solutions is established and the singularity of boundary blow-up solution is also studied for several cases including when $\frac{\rho (x,u(x))}{f(x,u(x))}\rightarrow 1$ as $x\rightarrow \partial \Omega $, which means that $\rho (x,u)$ is a large perturbation. We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of $d(x,\partial \Omega )$. Hence, the results of this paper are new even in the canonical case $p(\cdot )\equiv 2$. In particular, we do not have the comparison principle, because we don't make the monotone assumption of nonlinear term.

The so-called unified or Fokas method expresses the solution of an initial-boundary value problem (IBVP) for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple IBVPs, which will be referred to as problems of type~I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated IBVPs, which will be referred to as problems of type~II, there does not exist a classical transform pair and the solution cannot be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type~II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalized eigenfunctions described in the sixties by Gel'fand and his co-authors.

We study the asymptotic behavior of the solutions related to a family of singularly perturbed linear partial differential equations in the complex domain. The analytic solutions obtained by means of a Borel-Laplace summation procedure are represented by a formal power series in the perturbation parameter. Indeed, the geometry of the problem gives rise to a decomposition of the formal and analytic solutions so that a multi-level Gevrey order phenomenon appears. This result leans on a Malgrange-Sibuya theorem in several Gevrey levels.