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We consider a Cauchy problem for the inhomogeneous differential equation given in terms of an unbounded linear operator $A$ and the Caputo fractional derivative of order $\alpha \in (0, 2)$ in time. The previously known representation of…
In this paper, we consider the backward Cauchy problem of linear degenerate stochastic partial differential equations. We obtain the existence and uniqueness results in Sobolev space $L^p(\Omega; C([0,T];W^{m,p}))$ with both $m\geq 1$ and…
We study positive solutions of the pseudoparabolic equation with a sublinear source in $\mathbb{R}^n$. In this work, the source coefficient could be unbounded and time-dependent. Global existence of solutions to the Cauchy problem is…
In this paper, we study the global approximate multiplicative controllability for nonlinear degenerate parabolic Cauchy-Neumann problems. First, we will obtain embedding results for weighted Sobolev spaces, that have proved decisive in…
In this article, we consider an n-dimensional parabolic partial differential equation with a smooth coefficient term in the nonlinear gradient term. This equation was first introduced and analyzed in [E. Issoglio, On a non-linear…
We prove H\"older regularity results for a class of nonlinear parabolic problem with fractional-time derivative with nonlocal and Mittag-Leffler nonsingular kernel. Existence of weak solutions via approximating solutions is proved.…
We prove existence and uniqueness results for the time-dependent Hartree approximation arising in quantum dynamics. The Hartree equations of motion form a coupled system of nonlinear Schr{\"o}dinger equations for the evolution of product…
Motivated by models for biofilm growth, we consider Cauchy problems for quasilinear reaction diffusion equations where the diffusion coefficient has a porous medium type degeneracy as well as a singularity. We prove results on the…
We propose and analyse numerical schemes for a system of quasilinear, degenerate evolution equations modelling biofilm growth as well as other processes such as flow through porous media and the spreading of wildfires. The first equation in…
The paper is concerned with the Cauchy problem for a semi-linear hyperdissipative heat equation in Besov and Triebel-Lizorkin spaces which is related to the generalized Gauss-Weierstrass semi-group via Duhamel's principle. Using caloric…
We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal boundary condition under the assumption that…
We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau…
We consider mildly degenerate Kirchhoff equations with a small parameter and a weak dissipation term. We prove the existence of global solutions when the parameter is small with respect to the size of initial data. Then we provide…
We consider a degenerate wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary…
We establish the existence of weak solutions $u$ of the semilinear wave equation $\partial_t^2 u-\textrm{div}_x(a(t,x)\nabla_xu)=f_k(u)$ where $a(t,x)$ is equal to $1$ outside a compact set with respect to $x$ and a non-linear term $f_k$…
In this work, we analytically investigate a degenerating PDE system for phase separation and complete damage processes considered on a nonsmooth time-dependent domain with mixed boundary conditions. The evolution of the system is described…
The well known Duhamel's principle allows to reduce the Cauchy problem for linear inhomogeneous partial differential equations to the Cauchy problem for the corresponding homogeneous equation. In the paper one of the possible…
The existence and uniqueness of solutions of the Cauchy problem to a a stochastic parabolic integro-differential equation is investigated. The equattion considered arises in nonlinear filtering problem with a jump signal process and jump…
In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-\Delta)^\frac{\theta}{2} u=|x|^{-\gamma} u^p ,\quad x\in{\bf R}^N, t>0,…
Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of L_{p} spaces of functions whose regularity is defined by a Levy measure with O-regulary varying radial profile. Existence and uniqueness of a solution…