Related papers: Strictly Hyperbolic Cauchy Problems on $\mathbb{R}…
In this paper, we investigate the initial value problem for symmetric hyperbolic systems on globally hyperbolic Lorentzian manifolds with potentials that are both nonlocal in time and space. When the potential is retarded and uniformly…
We prove the well-posedness and regularity of solutions in mixed-norm weighted Sobolev spaces for a class of second-order parabolic and elliptic systems in divergence form in the half-space $\mathbb{R}^d_+ = \{x_d > 0\}$ subject to the…
We study the Dirichlet problem for first order hyperbolic quasi-linear functional PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. While the question of existence…
The initial value problem is well-defined on a class of spacetimes broader than the globally hyperbolic geometries for which existence and uniqueness theorems are traditionally proved. Simple examples are the time-nonorientable spacetimes…
In this paper, we consider the Cauchy problem for a hyperbolic equation $Q(\partial_t,\partial_x)u=0$ of any order $m\geq3$, where $t\geq0$ and $x\in\mathbb{R}^n$, and $Q=P_m+P_{m-1}+P_{m-2}$ is a sum of homogeneous hyperbolic polynomials…
We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical…
Consider a strictly hyperbolic $n\times n$ system of conservation laws, where each characteristic field is either genuinely nonlinear or linearly degenerate. In this standard setting, it is well known that there exists a Lipschitz semigroup…
In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless…
We study a class of parabolic equations having first order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton-Jacobi equation with superlinear Hamiltonian. We address the problem of having…
Systems of parabolic, possibly degenerate parabolic SPDEs are considered. Existence and uniqueness are established in Sobolev spaces. Similar results are obtained for a class of equations generalizing the deterministic first order symmetric…
We investigate the qualitative properties of the weak solutions to the boundary value problems for the hyperbolic fourth-order linear equations with constant coefficients in the plane bounded domain convex with respect to characteristics.…
In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we…
In the present article, a modified Cauchy problem (problem C) for the hyperbolic equation of the third order with the data on the equation's coefficients singularity plane is solved by Riemann method. The special class in which the solution…
It is studied the Cauchy problem for the equations of Burgers' type but with bounded dissipation flux. Such equations degenerate to hyperbolic ones as the velocity gradient tends to infinity. Thus the discontinuous solutions are permitted.…
Systems of PDEs comprised of a combination of constraints and evolution equations are ubiquitous in physics. For both theoretical and practical reasons, such as numerical integration, it is desirable to have a systematic understanding of…
We study a class of second-order degenerate linear parabolic equations in divergence form in $(-\infty, T) \times \mathbb R^d_+$ with homogeneous Dirichlet boundary condition on $(-\infty, T) \times \partial \mathbb R^d_+$, where $\mathbb…
We establish a mixed observability inequality for a class of degenerate hyperbolic equations on the cylindrical domain $\Omega = \mathbb{T} \times (0,1)$ with mixed Neumann Dirichlet boundary conditions. The degeneracy acts only in the…
The aim of the book is to present some recent results in the theory of stochastic It\^o equations with singular deterministic part (drift) and its applications to second-order elliptic and parabolic equations with singular first-order…
In this paper we study higher order weakly hyperbolic equations with time dependent non-regular coefficients. The non-regularity here means less than H\"older, namely bounded coefficients. As for second order equations in \cite{GR:14} we…
We study the large time behavior of solutions of first-order convex Hamilton-Jacobi Equations of Eikonal type set in the whole space. We assume that the solutions may have arbitrary growth. A complete study of the structure of solutions of…