On the nonexistence of global solutions of some initial-boundary value problems for the Korteweg-de Vries equation

We consider the problem of finite-time blow-up of solutions of a class of initial-boundary value problems for the Korteweg-de Vries equation. By using the method of optimal test functions corresponding to the boundary conditions, we obtain blow-up conditions for local (with respect to t > 0) solutions and estimate the blow-up time.     

An initial-boundary value problem with mixed lateral conditions for heat equation

Sunto Viene studiato un problema con condizioni laterali miste per l'equazione del calore, nel caso in cui la ? superficie di separazione ? abbia un punto caratteristico. Viene dato un teorema di esistenza e di unicità in spazi con peso a regolarità variabile.

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Entrata in Redazione l'8 marzo 1978.

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Partially supported by Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Bologna, Italy.     

On the structure of a solution set of controlled initial-boundary value problems

For a controlled nonlinear functional-operator equation of the Hammerstein type describing a wide class of controlled initial-boundary value problems, we obtain simple sufficient conditions for the convexity, pointwise boundedness and precompactness of the set of solutions (the reachability tube) in the Lebesgue space. As for boundedness and precompactness, we mean certain conditions of the majorant but not Volterra type requirements which give also the total (with respect to the whole set of admissible controls) preservation of solvability of mentioned equation. As some examples of reduction of a controlled initial-boundary (boundary) value problem to the equation under investigation, and verification of the proposed hypotheses for this equation, we consider the first initial-boundary value problem associated with a semilinear parabolic equation of the second order in a rather general form, and also the Dirichlet problem associated with a semilinear elliptic equation of the second order.     

On a class of initial-boundary value problems for equations of Korteweg-de Vries type

We consider a special class of initial-boundary value problems on the positive halfline x > 0 for the Korteweg-de Vries equation and its generalizations. For this class, we prove theorems on the nonexistence of global solutions for t > 0.     

On the convexity of global solvability sets for controlled initial-boundary value problems

We consider a controlled functional-operator equation that is a convenient form for describing of controlled initial-boundary value problems. For this equation, considered in a Banach ideal space, we define the set Ω of global solvability as the set of all admissible controls for which the equation has a global solution. We show that Ω is convex under the conditions imposed on the right-hand side of the equation. For each control in a given segment in Ω, we obtain a two-sided pointwise estimate for the corresponding solution under the abovementioned assumptions. We prove the theorem on the convex continuous dependence of the solution on the parameter specifying the displacement along the segment.     

On the blow-up of solutions to nonlinear initial-boundary value problems

We consider the problem of nonexistence (blow-up) of solutions of nonlinear evolution equations in the case of a bounded (with respect to the space variables) domain. Following the method of nonlinear capacity based on the application of test functions that are optimal (“characteristic”) for the corresponding nonlinear operators, we obtain conditions for the blowup of solutions to nonlinear initial-boundary value problems. We also show by examples that these conditions are sharp in the class of problems under consideration.     

Hermite methods for hyperbolic initial-boundary value problems

We study arbitrary-order Hermite difference methods for the numerical solution of initial-boundary value problems for symmetric hyperbolic systems. These differ from standard difference methods in that derivative data (or equivalently local polynomial expansions) are carried at each grid point. Time-stepping is achieved using staggered grids and Taylor series. We prove that methods using derivatives of order in each coordinate direction are stable under -independent CFL constraints and converge at order . The stability proof relies on the fact that the Hermite interpolation process generally decreases a seminorm of the solution. We present numerical experiments demonstrating the resolution of the methods for large as well as illustrating the basic theoretical results.

     

A remark on the theory of initial-boundary value problems for the equation of rods and beams

Initial value problems are studied for the beam equation with different conditions at the endpoints. Theorems are proved on the uniqueness, existence, and stability of the posed problems in the classes of regular and generalized solutions.     

Solvability of the initial-boundary value problem for an integrodifferential equation

Under study is the well-posedness of the Cauchy problem for the nonstationary radiation transfer equation with generalized matching conditions at the interface between the media. We prove the existence of a unique strongly continuous semigroup of resolvents, estimate its order of growth, and consider the question of stabilization of the nonstationary solution.     

Reduction of the basic initial-boundary value problems for the Navier-Stokes equations to initial-boundary value problems for nonlinear parabolic systems of pseudodifferential equations

We consider initial-boundary value problems for systems of Navier-Stokes equations, prescribing on the boundary the velocities, the stresses or the normal component of the velocity and the tangential stresses. We show that they can be reduced to initial-boundary value problems for systems of the form v_t + Av + Kv=f, where A is a linear elliptic operator, containing a nonlocal term, while K is a nonlinear operator. For these problems we prove a local solvability theorem in the Sobolev-Slobodetskii spaces W ₂ ^{, /2} .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 171, pp. 36–52, 1989.