Related papers: On coupled systems of PDEs with unbounded coeffici…

We study generalized solutions of an evolutionary equation related to a densely defined skew-symmetric operator in a real Hilbert space. We establish existence of a contractive semigroup, which provides generalized solutions, and find…

For a control Cauchy problem $$\dot x= {f}(t,x,u,v) +\sum_{\alpha=1}^m g_\alpha(x) \dot u_\alpha,\quad x(a)=\bar x, $$ on an interval $[a,b]$, we propose a notion of limit solution $x,$ verifying the following properties: i) $x$ is defined…

This work is focused on establishing sufficient conditions to guarantee the well-posedness of the following nonlinear fractional semidiscrete model \begin{equation*} \begin{cases} \mathbb D^\beta_t u(n,t)= B u(n,t) + f(n-ct,u(n,t)),\,…

The existence and uniqueness in Sobolev spaces of solutions of the Cauchy problem to parabolic integro-differential equation of the order {\alpha}\in(0,2) is investigated. The principal part of the operator has kernel…

Let $p:\R\to(1,\infty)$ be a globally log-H\"older continuous variable exponent and $w:\R\to[0,\infty]$ be a weight. We prove that the Cauchy singular integral operator $S$ is bounded on the weighted variable Lebesgue space…

The family of Cauchy transforms \[C_{g}(z,w) = -\frac{1}{\pi}\int_{\mathbb{C} } \frac{g(u)}{\overline{u-w} (u-z) } da(u ),\] where the measurable function $g$ with compact (essential) support satisfies $0 \leq g\leq 1,$ and suitably defined…

For hyperbolic differential operators $P$ with non-effectively hyperbolic double characteristics, we study the relationship between the Gevrey well-posedness threshold for strong well-posedness and the associated Hamilton map and flow. In…

A system of two operator equations is considered - one of pseudomonotone type and the other of strongly monotone type - both being strongly coupled. Conditions are given that allow to reduce the solvability of this system to a single…

We consider the Cauchy problem for one-dimensional p-system with damping of space-dependent coefficient. This system models the compressible flow through porous media in the Lagrangean coordinate. Our concern is an asymptotic behavior of…

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of…

In this paper, we study the following Cauchy problem for linear visco-elastic damped wave models with a general time-dependent coefficient $g=g(t)$: \begin{equation} \label{EqAbstract} \tag{$\star$} \begin{cases} u_{tt}- \Delta u +…

In this work, we prove existence and uniqueness of a bounded viscosity solution for the Cauchy problem of degenerate parabolic equations with variable exponent coefficients. We construct the solution directly using the stochastic…

We study a nonlinear porous medium type equation involving the infinity Laplacian operator. We first consider the problem posed on a bounded domain and prove existence of maximal nonnegative viscosity solutions. Uniqueness is obtained for…

We consider the degenerate parabolic equation $$ \partial_t u +\mathrm{div} {\mathfrak f}_{\bf x}(u)=\mathrm{div}(\mathrm{div} ( A_{\bf x}(u) ) ), \ \ {\bf x} \in M, \ \ t\geq 0 $$ on a smooth, compact, $d$-dimensional Riemannian manifold…

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, consider a self-adjoint matrix second order elliptic differential operator $\mathcal{B}_\varepsilon$, $0<\varepsilon \leqslant 1$. The principal part of the operator is given in a factorised form, the…

We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameters $p>2$ and $s\in (0,1)$ (fractional exponent). We show that the Cauchy Problem for data in the Lebesgue $L^q$ spaces…

We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho…

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial-boundary problem (P) for the nonlinear diffusion equation in an unbounded domain $\Omega\subset\mathbb{R}^N$…

We consider the Cauchy problem for stochastic fractional evolution equations with Caputo time fractional derivative of order $1<\alpha<2$ and space variable coefficients on an unbounded domain. The space derivatives that appear in the…

Parabolic integro-differential non degenerate Cauchy problem is considered in the scale of H\"older spaces of functions whose regularity is defined by a radially O-regularly varying L\'evy measure. Existence and uniqueness and the estimates…