Smoothness of solutions of second-order hyperbolic equations with discontinuous coefficients

Translated from Matematicheskie Zametki, Vol. 49, No. 6, pp. 3–11, June, 1991.     

Nonlocal problem for complete second-order hyperbolic operator-differential equations with variable domains

We prove the well-posed solvability (in the strong sense) of complete second-order hyperbolic operator-differential equations with variable domains of unbounded operator coefficients under nonlocal initial conditions. We are the first to establish the well-posed solvability of the mixed problem for the complete string vibration equation with nonstationary boundary conditions and nonlocal initial conditions.     

Goursat problem for two-dimensional second-order hyperbolic operator-differential equations with variable domains

We develop a modification of the energy inequality method and use it to prove the well-posedness of the Goursat problem for linear second-order hyperbolic differential equations with operator coefficients whose domains depend on the two-dimensional time. An energy inequality for strong solutions of this Goursat problem is derived with the help of abstract smoothing operators, and we prove that the range of the problem is dense by using properties of a regularizing Cauchy problem whose inverse operator is a family of smoothing operators of a new type. We give an example of a well-posed boundary value problem for a two-dimensional complete second-order hyperbolic partial differential equation with Goursat conditions and with a boundary condition depending on the two-dimensional time.     

On Stokes operators with variable viscosity in bounded and unbounded domains

We consider a generalization of the Stokes resolvent equation, where the constant viscosity is replaced by a general given positive function. Such a system arises in many situations as linearized system, when the viscosity of an incompressible, viscous fluid depends on some other quantities. We prove that an associated Stokes-like operator generates an analytic semi-group and admits a bounded H _∞ -calculus, which implies the maximal L ^q -regularity of the corresponding parabolic evolution equation. The analysis is done for a large class of unbounded domains with -boundary for some r > d with r ≥ q, q′. In particular, the existence of an L ^q -Helmholtz projection is assumed.     

Euler-Poisson-Darboux differential-operator equation with variable domains of smooth operators

We prove the strong well-posed solvability of the Cauchy problem for a second-order singular hyperbolic differential equation with variable domain of variable unbounded operator coefficients and for the mixed problem for a complete equation of string vibrations with a strong singularity in time and with a time-dependent boundary condition.     

First-order differential-operator equation with variable domains of piecewise smooth operators

We prove the existence, uniqueness, and smoothness of weak solutions of a first-order differential-operator equation with variable domains of nonself-adjoint piecewise smooth operators for which one has the corresponding majorant operators. We analyze the well-posedness and smoothness of weak solutions of three new mixed problems with piecewise smooth (in time) coefficients in the equations of finite and infinite order and in the boundary conditions.     

L p -theory for second-order elliptic operators with unbounded coefficients

Second-order elliptic operators with unbounded coefficients of the form ${Au := -{\rm div}(a\nabla u) + F . \nabla u + Vu}$ in ${L^{p}(\mathbb{R}^{N}) (N \in \mathbb{N}, 1 < p < \infty)}$ are considered, which are the same as in recent papers Metafune et?al. (Z Anal Anwendungen 24:497–521, 2005), Arendt et?al. (J Operator Theory 55:185–211, 2006; J Math Anal Appl 338: 505–517, 2008) and Metafune et?al. (Forum Math 22:583–601, 2010). A new criterion for the m-accretivity and m-sectoriality of A in ${L^{p}(\mathbb{R}^{N})}$ is presented via a certain identity that behaves like a sesquilinear form over L ^p ×?L ^p'. It partially improves the results in (Metafune et?al. in Z Anal Anwendungen 24:497–521, 2005) and (Metafune et?al. in Forum Math 22:583–601, 2010) with a different approach. The result naturally extends Kato’s criterion in (Kato in Math Stud 55:253–266, 1981) for the nonnegative selfadjointness to the case of p ≠?2. The simplicity is illustrated with the typical example ${Au = -u\hspace{1pt}'' + x^{3}u\hspace{1pt}' + c |x|^{\gamma}u}$ in ${L^p(\mathbb{R})}$ which is dealt with in (Arendt et?al. in J Operator Theory 55:185–211, 2006; Arendt et?al. in J Math Anal Appl 338: 505–517, 2008).     

Degenerated nonlinear hyperbolic equation with discontinuous coefficients

The partial differential equations with discontinuous coefficients have been extended in various directions by a number of authors [2], [3], [11], [13], [14], [16]. This paper deals with a mixed problem for a degenerated nonlinear hyperbolic equation with discontinuous coefficients.     

Global regular solutions for the 3D Kawahara equation posed on unbounded domains

An initial boundary value problem for the 3D Kawahara equation posed on a channel-type domain was considered. The existence and uniqueness results for global regular solutions as well as exponential decay of small solutions in the H²-norm were established.     

Existence theorem for a stochastic wave equation with discontinuous unbounded coefficients

We prove an existence theorem for stochastic hyperbolic equations with measurable locally bounded coefficients. A solution of a stochastic hyperbolic equation is understood as a martingale solution of the stochastic inclusion corresponding to the equation.     

Classical solution of the first mixed problem for second-order hyperbolic equation in curvilinear half-strip with variable coefficients

The classical solution of the first mixed problem for a second-order hyperbolic equation with variable coefficients in the case of two independent variables in a curvilinear half-strip is considered. The existence and uniqueness of the classical solution under specific smoothness and matching conditions for given functions are proved. A method is proposed for constructing the solution using the method of sequential approximations for a system of integral equations of the second kind.