Related papers: Parabolic equations involving Bessel operators and…

We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional…

We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or $L^1$ data. The key difficulty consists in a presence of a monotone operator~$A$ subjected to a non-standard growth condition,…

The purpose of this paper is to further exemplify an approach to evolutionary problems originally developed in earlier works for a special case and later extended to more general evolutionary problems. We are here concerned with the $(1+1)$…

We study the uniqueness of non-negative solutions of the equation \begin{align*} \partial_t\left(|u|^{p-2}u\right)\,=\, \operatorname{div}(|\nabla u|^{p-2}\nabla u). \end{align*} Basic estimates are derived with the Galerkin Method.

We solve the Cauchy problem defined by the fractional partial differential equation $[\partial_{tt}-\kappa\mathbb{D}]u=0$, with $\mathbb{D}$ the pseudo-differential Riesz operator of first order, and the initial conditions…

In this paper we study a doubly degenerate parabolic equation involving a convection term and the operator $\mathcal{A}_\mu u:=-\Delta_p u +\mu (-\Delta)^s_q u$ which is a linear combination of the $p$-Laplacian and the fractional…

In this paper we consider fractional quasi-Bessel equations $$\sum_{i=1}^{m}d_i x^{\alpha_i+p_i}D^{\alpha_i} u(x) + (x^\beta - \nu^2)u(x)=0$$ and construct their existence and uniqueness theory in the class of fractional series. Our…

This paper is concerned with the strong solution to the Cauchy-Dirichlet problem for backward stochastic partial differential equations of parabolic type. Existence and uniqueness theorems are obtained, due to an application of the…

We consider solutions of the Cauchy problem for semilinear equations with (possibly) different L\'evy operators. We provide various results on their convergence under the assumption that symbols of the involved operators converge to the…

We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic…

We study the solutions $u=u(x,t)$ to the Cauchy problem on $\mathbb Z^d\times(0,\infty)$ for the parabolic equation $\partial_t u=\Delta u+\xi u$ with initial data $u(x,0)=1_{\{0\}}(x)$. Here $\Delta$ is the discrete Laplacian on $\mathbb…

We study the following quasilinear partial differential equation with two subdifferential operators: $${\frac{\partial u}{\partial s}(s,x)} + (\mathcal{L}u)(s,x,u(s,x),(\nabla u(s,x))^\ast\sigma(s,x,u(s,x))) + f(s,x,u(s,x),(\nabla…

In the present paper, we prove time decay estimates of solutions in weighted Sobolev spaces to the second order evolution equation with fractional Laplacian and damping for data in Besov spaces. Our estimates generalize the estimates…

We consider the Cauchy problem of the semilinear wave equation with a damping term \begin{align*} u_{tt} - \Delta u + c(t,x) u_t = |u|^p, \quad (t,x)\in (0,\infty)\times \mathbb{R}^N,\quad u(0,x) = \varepsilon u_0(x), \ u_t(0,x) =…

We consider an evolution equation with the regularized fractional derivative of an order $\alpha \in (0,1)$ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables.…

We prove well-posedness in weighted tent spaces of weak solutions to the Cauchy problem $\partial_t u - \mathrm{div} A \nabla u = f, u(0)=0$, where the source $f$ also lies in (different) weighted tent spaces, provided the complex…

We consider the ill-posed Cauchy problem for the polyharmonic heat equation on recovering a function, satisfying the equation $(\partial _t + (- \Delta)^m) u=0$ in a cylindrical domain in the half-space ${\mathbb R}^n \times [0,+\infty)$,…

We investigate the following quasilinear parabolic and singular equation, {equation} \tag{{\rm P$_t$}} \{{aligned} & u_t-\Delta_p u =\frac{1}{u^\delta}+f(x,u)\;\text{in}\,(0,T)\times\Omega, & u =0\,\text{on}…

In this paper we study the Cauchy problem for new classes of parabolic type pseudodifferential equations over the rings of finite adeles and adeles. We show that the adelic topology is metrizable and give an explicit metric. We find…

We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, where the exponent satisfies the doubling condition. In particular, both the so called logconvex and…