Existenzsätze mit der linienmethode für parabolische probleme und periodische lösungen von ut=f(t,x,u,ux,uxx)

The (longitudinal) method of lines transforms a parabolic equation into a first order system of ordinary differential equations by discretization of the spatial variable. It is shown how to obtain existence theorems for nonlinear parabolic equations from those for ordinary differential equations under general growth conditions and weak regularity assumptions. The method is demonstrated in proving a new existence theorem for periodic solutions to u_t=f(t,x,u,u_x,u_xx) with boundary conditions of Dirichlet type.     

Local solvability in c∞ for quasi-linear weakly hyperbolic equations of second order

We consider here the initial value problem for quasi-linear weakly hyperbolic equations of type u_tt - a(t,x,u)u_xx = ?(t,x,u). Assuming the degeneracy of the elliptic term has finite order with respect to u, the local solvability in the class of smooth functions is proved.     

On the propagation of perturbations for essentially nonautonomous quasilinear first-order equations

This paper deals with generalized solutions of the Cauchy problem for the equationu _t + [A(t, x, u)] _x +B(t, x, u) = 0 (t, x) ∈ ℝ₊ × ℝ. Here A, B may depend essentially on t, x; for example, they may tend to zero or to infinity as t becomes infinite. Sufficient conditions are obtained for the presence and the absence of finite time extinction and space localization. These phenomena have been studied earlier mainly for degenerate parabolic equations. In the case of first-order equations the situation is more complicated due to the discontinuity of solutions. The essential dependence of the coefficients on t, x gives rise to a threshold phenomenon: the presence of the finite time extinction depends on the maximum of the modulus of the initial function. Bibliography: 29 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 89–117, 1994.     

Painlevé Classification of All Semilinear Partial Differential Equations of the Second Order. II. Parabolic and Higher Dimensional Equations

This paper extends the work of the previous paper (I) on the Painlevé classification of second-order semilinear partial differential equations to the case of parabolic equations in two independent variables, u_xx = F(x, y, u, u_x, u_y), and irreducible equations in three or more independent variables of the form, Σ_ijR_ij (x₁,…, x_n)u,_ij = F(x₁,…, x_n; u,₁,…, u,_n). In each case, F is assumed to be rational in u and its first derivatives and no other simplifying assumptions are made. In addition to the 22 hyperbolic equations found in paper I, we find 10 equivalence classes of parabolic equations with the Painlevé property, denoted PS-I, PS-I1,…, PS-X, equation PS-II being a generalization of Burgers' equation denoted the Forsyth-Burgers equation, and 13 higher-dimensional Painlevé equations, denoted GS-I, GS-II,…, GS-XIII. The lists are complete up to the equivalence relation of Möbius transformations in u and arbitrary changes of the independent variables. In order to avoid repetition, the proofs are sketched very briefly in cases where they closely resemble those for the corresponding hyperbolic problem. Every equation is solved by transforming to a linear partial differential equation, from which it follows that there are no non trivial soliton equations among the two classes of Painlevé equations treated in this paper.     

Fourth-order finite difference scheme for a system of quasilinear elliptic equations

The system of two quasilinear elliptic equations is approximated by the method of lines, which has the truncation error O(h²) at points neighboring the boundary and O(h⁴) at the most interior points. It is proved that the global error of the method is O(h⁴) at all mesh points. The two-point boundary value problem for the system of ordinary differential equations that arises from the method of lines is solved by the O(h⁴) convergent finite difference scheme, suitable to the equations of the form u_xx = f(x, u) without the first derivative u_x. The system of algebraic equations obtained by the full discretization is solved by Gauss elimination method for three diagonal matrices combined with the method of iterations. A numerical example is presented.     

Strong Solvability of a Unilateral Boundary Value Problem for Nonlinear Parabolic Operators

Strong solvability in Sobolev spaces is proved for a unilateral boundary value problem for nonlinear parabolic operators. The operator is assumed to be of Carathéodory type and to satisfy a suitable ellipticity condition; only measurability with respect to the independent variable X is required. The main tools of the proof are an estimate for the second derivatives of functions which satisfy the unilateral boundary conditions and the monotonicity of the operator − u _t with respect to Δu for the same functions.     

On the asymptotic behavior of solutions of a mildly nonlinear parabolic system

In this paper we study the asymptotic behavior of solutions to the mixed initial boundary value problem for the system of nonlinear parabolic equations

$\text{?u}{}_{\mathrm{t}}\text{+Lu=f(x.t.u.v)}\text{?v}{}_{\mathrm{t}}\text{+Mv=g(x,t,u,v)}$

We show, under suitable technical assumptions, that these solutions converge to solutions of the Dirichlet problem for the corresponding limiting elliptic system, provided that the solution of the Dirichlet problem is unique.     

The cauchy problem of a degenerate parabolic equation

This note is concerned with the existence of a weak solution for a degenerate Cauchy problem of parabolic type in then-dimensional spaceR ⁿ. The degenerate property is in the sense that the matrix (a _ij(t,x)) involved in the differential operator is not necessarily positive definite. The essential idea is the construction of a suitable function spaceH and to prove the existence of a weak solution inH.     

Parabolic equations with nonlocal nonlinear source

Conclusion The results presented in § 1 and § 2 can serve as a basis for further study of equations like (1.1), (2.1)–(2.3). For example, using the obtained estimates for a solution u(x,t) together with the well-known estimates for solutions to the Cauchy problem or the maximum principle for parabolic equations [6, 7], we can easily obtain estimates for the derivativesu _t (x, t),u _tt (x, t), etc., as well as estimates for the derivatives with respect to the space variables.Concluding the article, we note that, in our opinion, together with the questions of existence and nonexistence of smooth solutions it is worthwhile to study some questions that concern qualitative properties of solutions to the considered equations, for example the questions of localization of solutions and some other questions.Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 100–1005, September–October, 1994.     

Differential transformations of parabolic second-order operators in the plane

Darboux’s classical results about transformations of second-order hyperbolic equations by means of differential substitutions are extended to the case of parabolic equations of the form Lu = (D _x ² + a(x, y)D _x + b(x, y)D _y + c(x, y))u = 0. We prove a general theorem that provides a way to determine admissible differential substitutions for such parabolic equations. It turns out that higher order transforming operators can always be represented as a composition of first-order operators that define a series of consecutive transformations. The existence of inverse transformations imposes some differential constrains on the coefficients of the initial operator. We show that these constraints may imply famous integrable equations, in particular, the Boussinesq equation.     

On viscosity solutions for nontotally parabolic fully nonlinear equations

It is shown how to prove global unique solvability of the first initial-boundary value problem in the class of continuous viscosity solutions for some classes of equations −u_t+F(u_x,u_xx)=g(x, t, u_x), where F(p, A) is elliptic only on some nonlinear subsets of values of the arguments (p, A). For this purpose we use the techniques developed in the theory of viscosity solutions for degenerate elliptic equations. Bibliography: 12 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 233, 1996, pp. 112–130.     

An inverse problem for an equation of parabolic type

In this paper we consider an inverse problem for the differential equationu _t =u _xx +q(x, t) u; the problem amounts to finding the coefficient q(x, t) from the solution of a series of Cauchy problems for this equation, the solution being specified on some manifold. Our main result is a proof of a uniqueness theorem.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 595–600, April, 1976.     

Smoluchowski-Kramers approximation for a general class of SPDEs

We prove that the so-called Smoluchowski-Kramers approximation holds for a class of partial differential equations perturbed by a non-Gaussian noisy term. Namely, we show that the solution of the one-dimensional semi-linear stochastic damped wave equations , u(0) = u₀, u_t (0) = v₀, endowed with Dirichlet boundary conditions, converges as the parameter μ goes to zero to the solution of the semi-linear stochastic heat equation , u(0) = u₀, endowed with Dirichlet boundary conditions. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Determination of a control function in three‐dimensional parabolic equations by Legendre pseudospectral method

A Legendre pseudospectral method is proposed for solving approximately an inverse problem of determining an unknown control parameter p(t) which is the coefficient of the solution u(x, y, z, t) in a diffusion equation in a three‐dimensional region. The diffusion equation is to be solved subject to suitably prescribed initial‐boundary conditions. The presence of the unknown coefficient p(t) requires an extra condition. This extra condition considered as the integral overspecification over the spacial domain. For discretizing the problem, after homogenization of the boundary conditions, we apply the Legendre pseudospectral method in a matrix based manner. As a results a system of nonlinear differential algebraic equations is generated. Then by using suitable transformation, the problem will be converted to a homogeneous time varying system of linear ordinary differential equations. Also a pseudospectral method for efficient solving of the resulted system of ordinary differential equations is proposed. The solution of this system gives the approximation to values of u and p. The matrix based structure of the present method makes it easy to implement. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed computational procedure. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 74‐93, 2012     

Weak solutionsu(x,t) to parabolic partial differential equations with coefficients that depend uponu(yl, σ, (t,u(x,t))), 1 = 1, … k

The existence of weak solutionsu(x, t) to parabolic partial differential equations with coefficients that depend onu(y_l, σ_l(t, u(x, t))), l = 1,… k, is demonstrated using a retardation of the time arguments in the coefficients along with regularity and compactness results for solutions of linear parabolic partial differential equations.     

On a forward-backward parabolic equation

Boundary value problems for the equation $$\operatorname{sgn} (x) \cdot u_y - u_{xx} + ku = f(x,y)$$ (where k is a positive constant and ? is a given function) are investigated. The domain of the solutions will be the whole upper half-plane y>0, or the half-plane y>0 cut along the positive y-axis. We are interested in square integrable solutions u, with square integrable generalized derivatives u_y and u_xx. Existence theorems are proved, with an integral equations technique. Thus a theory is developed of Wiener-Hopf integral equations of the first kind with solutions belonging to Sobolev spaces.     

On the Stabilization of a Solution of the Cauchy Problem for One Class of Integro-Differential Equations

We consider a solution of the Cauchy problem u(t, x), t > 0, x ∈ R ², for one class of integro-differential equations. These equations have the following specific feature: the matrix of the coefficients of higher derivatives is degenerate for all x. We establish conditions for the existence of the limit lim _t→∞ u(t, x) = v(x) and represent the solution of the Cauchy problem in explicit form in terms of the coefficients of the equation.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1699 – 1706, December, 2004.     

On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach

The classic problem of regularity of boundary points for higher-order partial differential equations (PDEs) is concerned. For second-order elliptic and parabolic equations, this study was completed by Wiener’s (J. Math. Phys. Mass. Inst. Tech. 3:127–146, 1924) and Petrovskii’s (Math. Ann. 109:424–444, 1934) criteria, and was extended to more general equations including quasilinear ones. Since the 1960–1970s, the main success was achieved for 2mth-order elliptic PDEs; e.g., by Kondrat’ev and Maz’ya. However, the higher-order parabolic ones, with infinitely oscillatory kernels, were not studied in such details. As a basic model, explaining typical difficulties of regularity issues, the 1D bi-harmonic equation in a domain shrinking to the origin (0, 0) is concentrated upon:

An O(k2 + kh2 + h4) arithmetic average discretization for the solution of 1‐D nonlinear parabolic equations

This article develops a new two‐level three‐point implicit finite difference scheme of order 2 in time and 4 in space based on arithmetic average discretization for the solution of nonlinear parabolic equation ε u_xx = f(x, t, u, u_x, u_t), 0 < x < 1, t > 0 subject to appropriate initial and Dirichlet boundary conditions, where ε > 0 is a small positive constant. We also propose a new explicit difference scheme of order 2 in time and 4 in space for the estimates of (?u/?x). The main objective is the proposed formulas are directly applicable to both singular and nonsingular problems. We do not require any fictitious points outside the solution region and any special technique to handle the singular problems. Stability analysis of a model problem is discussed. Numerical results are provided to validate the usefulness of the proposed formulas. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Convergence Rate of Galerkin Method for a Certain Class of Nonlinear Operator-Differential Equations

In this article, we study a Galerkin method for a nonstationary operator equation with a leading self-adjoint operator A(t) and a subordinate nonlinear operator F. The existence of the strong solutions of the Cauchy problem for differential and approximate equations are proved. New error estimates for the approximate solutions and their derivatives are obtained. The developed method is applied to an initial boundary value problem for a partial differential equation of parabolic type.