A fundamental solution to the Cauchy problem for a fourth-order pseudoparabolic equation

The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.     

Cauchy problem for quasilinear hyperbolic systems of shallow water equations

This article investigates the Cauchy problem for two different models (modified and classical), governed by quasilinear hyperbolic systems that arise in shallow water theory. Under certain reasonable hypotheses on the initial data, we obtain the global smooth solutions for both the systems. The bounds on simple wave solutions of the modified system are shown to depend on the parameter H characterizing the advective transport of impulse. Similarly the bounds on simple wave solutions of the classical system describing the flow over a sloping bottom with profile b(x) are shown to depend on the bottom topography. On the other hand, if the initial data are specified differently, then it is shown that solutions for both the systems exhibit finite time blow-up from specific smooth initial data. Moreover, we show that an increase in H and convexity of b would reduce the time taken for the solutions to blow up.     

On the mixed problem for quasilinear pseudoparabolic equation

In this paper the multidimensional mixed problem for the quasilinear pseudoparabolic equation u_t-L_xu-εL_xu_t=f(t,x,u) is considered. L_x is a differential operator, which composes (with boundary operator) a self adjoint operator. An existence, uniqueness and also continuous dependense on the small parameter ε>0 of generalized solution is proved. The estimation of the difference of exact and approximate solutions is obtained     

Uniqueness and stability of solution for Cauchy problem of degenerate quasilinear parabolic equations

The uniqueness and existence of BV-solutions for Cauchy problem of the form are proved.     

The generalized Cauchy problem with data on two surfaces for a quasilinear analytic system

We consider the generalized Cauchy problem with data on two surfaces for a second-order quasilinear analytic system. The distinction of the generalized Cauchy problem from the traditional statement of the Cauchy problem is that the initial conditions for different unknown functions are given on different surfaces: for each unknown function we pose its own initial condition on its own coordinate axis. Earlier, the generalized Cauchy problem was considered in the works of C. Riquier, N. M. Gyunter, S. L. Sobolev, N. A. Lednev, V. M. Teshukov, and S. P. Bautin. In this article we construct a solution to the generalized Cauchy problem in the case when the system of partial differential equations additionally contains the values of the derivatives of the unknown functions (in particular outer derivatives) given on the coordinate axes. The last circumstance is a principal distinction of the problem in the present article from the generalized Cauchy problems studied earlier.     

Higher approximations of the averaging method for quasilinear parabolic equations with fast oscillating principal part in the case of the Cauchy problem

An algorithm for constructing the higher approximations of the averaging method for quasilinear parabolic equations with fast oscillating coefficients is suggested and justified. Translated fromMatematicheskie Zametki, Vol. 65, No. 4, pp. 562–572, April, 1999.     

The Cauchy problem for a class of quasilinear parabolic equations

The present paper is concerned with the Cauchy problem for the parabolic equation u_t+H(t,x,u,u)=u. New conditions guaranteeing the global classical solvability are formulated. Moreover, it is shown that the same conditions guarantee the global existence of the Lipschitz continuous viscosity solution for the related Hamilton–Jacobi equation. Mathematics Subject Classification (2000) 35K15, 35F25     

The Cauchy problem for quasilinear SG‐hyperbolic systems

We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) ∈ [0, T ] × ?ⁿ and presenting a linear growth for |x | → ∞. We prove well‐posedness in the Schwartz space ?? (?ⁿ ). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cauchy problem for integrable discrete equations on quad-graphs

Initial value problems for the integrable discrete equations on quad-graphs are investigated. We give a geometric criterion of when such a problem is well-posed. In the basic example of the discrete KdV equation an effective integration scheme based on the matrix factorization problem is proposed and the interaction of the solutions with the localized defects in the regular square lattice are discussed in details. The examples of kinks and solitons on various quad-graphs, including quasiperiodic tilings, are presented.Dedicated to S. P. Novikov on his 65 birthdayOn leave from Landau Institute for Theoretical Physics, Chernogolovka, Russia.     

The Cauchy problem for the system of thermoelasticity equations in space

We consider the problem of analytic continuation of the solution of the system of thermoelasticity equations in a bounded three-dimensional domain on the basis of known values of the solution and the corresponding stress on a part of the boundary, i.e., the Cauchy problem. We construct an approximate solution of the problem based on the method of Carleman's function.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 212–217, August, 1998.In conclusion, the authors wish to thank Professor M. M. Lavrent'ev and Professor Sh. Ya. Yarmukhamedov for setting the problem and for discussions in the course of the solution.     

On analytic solutions to the generalized Cauchy problem with data on three surfaces for a quasilinear system

The generalized Cauchy problem with data on three surfaces is under consideration for a quasilinear analytic system of the third order. Under some simplifying assumption, we find necessary and sufficient conditions for existence of a solution in the form of triple series in the powers of the independent variables. We obtain convenient sufficient conditions under which the data of the generalized Cauchy problem has a unique locally analytic solution. We give counterexamples demonstrating that in the case we study it is impossible to state necessary and sufficient conditions for analytic solvability of the generalized Cauchy problem. We also show that the analytic solution can fail to exist even if the generalized Cauchy problem with data on three surfaces has a formal solution since the series converge only at a sole point, the origin.     

On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order

We study the wellposedness in the Gevrey classes G^s and in C^∞ of the Cauchy problem for weakly hyperbolic equations of higher order. In this paper we shall give a new approach to the case that the characteristic roots oscillate rapidly and vanish at an infinite number of points. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence of weak discontinuous solutions to the Cauchy problem with small BV initial data for quasilinear hyperbolic systems

In this paper, we study the Cauchy problem for quasilinear hyperbolic system with a kind of non‐smooth initial data. Under the assumption that the initial data possess a suitably small bounded variation norm, a necessary and sufficient condition is obtained to guarantee the existence and uniqueness of global weak discontinuous solution. Copyright © 2014 John Wiley & Sons, Ltd.     

On the Cauchy problem of degenerate hyperbolic equations

In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.

     

Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for a nonhomogeneous quasilinear hyperbolic system

It is proven that if the leftmost eigenvalue is weakly linearly degenerate, then the Cauchy problem for a class of nonhomogeneous quasilinear hyperbolic systems with small and decaying initial data given on a semi-bounded axis admits a unique global C¹ solution on the domain , where x=xn(t) is the fastest forward characteristic emanating from the origin. As an application of our result, we prove the existence of global classical, C¹ solutions of the flow equations of a model class of fluids with viscosity induced by fading memory with small smooth initial data given on a semi-bounded axis.     

Well-posedness for the Cauchy problem of coupled Hirota equations with low regularity data

The local well-posedness of the Cauchy problem for the coupled Hirota equations is established in Sobolev spaces Hs×Hs by the Fourier restriction norm method.     

Explicit construction of measure solutions of Cauchy problem for transportation equations

The transportation equations are a mathematical model of zero-pressure flow in gas dynamics and the adhesion particle dynamics system to explain the formation of large scale structures in the universe. With the help of convex hull of a potential function, the solution is explicitly constructed here. It is straightforward to prove that the solution is a global measure one. And Dirac delta-shocks explained as the concentration of particles may develop in the solution. Project supported by the Institute of Mathematics, Chinese Academy of Sciences and by the National Fundamental Research Program of State Commission of Science and Technology of China, and Chinese Academy of Sciences.     

Potential well method for Cauchy problem of generalized double dispersion equations

In this paper we study Cauchy problem of generalized double dispersion equations utt−uxx−uxxtt+uxxxx=f(u)_xx, where f(u)=p|u|, p>1 or u2k, . By introducing a family of potential wells we not only get a threshold result of global existence and nonexistence of solutions, but also obtain the invariance of some sets and vacuum isolating of solutions. In addition, the global existence and finite time blow up of solutions for problem with critical initial conditions E(0)=d, I(u₀)?0 or I(u₀)<0 are proved.