On A.I. Koshelev’s work on regularity theory of generalized solutions of quasilinear systems of elliptic and parabolic type

This is a survey of A.I. Koshelev’s studies in the theory of regular solutions of boundary value problems, based on iterative processes converging in both the energy norm and the strong norm as well as on a priori estimates in weighted function spaces. In numerous cases, Koshelev’s estimates contain explicitly computable and sometimes sharp (unimprovable) constants. The results obtained for a broad class of problems were adapted by Koshelev to the study of boundary value problems of nonlinear elasticity and problems of hydrodynamics of viscous fluids.     

Hölder estimates for quasilinear doubly degenerate parabolic equations

For positive bounded generalized solutions of degenerate parabolic equations of the form t 0, m 2, one establishes local Hölder estimates, independent of the lower bounds of the indicated solutions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 171, pp. 70–105, 1989.     

Asymptotic stability of some quasilinear parabolic equations in divergence form. L∞ results

In this paper we study the behavior of solutions of some quasilinear parabolic equations of the form

$\text{(}\text{?u}\text{?t}\text{) ?}\text{∑}\text{i=1}\text{n}\text{(}\text{d}\text{dx}{}_{\mathrm{i}}\text{) a}{}_{\mathrm{i}}\text{(x, t, u, u}{}_{\mathrm{x}}\text{) + a(x, t, u, u}{}_{\mathrm{x}}\text{)u + f(x, t) = O,}$

as t → ∞. In particular, the solutions of these equations will decay to zero as t → ∞ in the L^∞ norm.     

Finite-time blow-up for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear degenerate Keller–Segel systems of parabolic–parabolic type in a ball of ${\mathbb{R}}^N$ ($N\ge 2$). In the case of non-degenerate diffusion, Cie?lak–Stinner [3], [4] proved that if $q>m+\frac{2}{N}$, where m denotes the intensity of diffusion and q denotes the nonlinearity, then there exist initial data such that the corresponding solution blows up in finite time. As to the case of degenerate diffusion, it is known that a solution blows up if $q>m+\frac{2}{N}$ (see Ishida–Yokota [13]); however, whether the blow-up time is finite or infinite has been unknown. This paper gives an answer to the unsolved problem. Indeed, the finite-time blow-up of energy solutions is established when $q>m+\frac{2}{N}$.     

Phragmen-Lindelöf type results for second order quasilinear parabolic equations in ℝ2

In this paper Phragmen-Lindelöf type growth-decay estimates are derived for solutions of initial-boundary value problems associated with a class of quasilinear parabolic equations defined on a semi-infinite strip in ². The particular problems considered are ones in which homogeneous initial data and homogeneous Dirichlet conditions on the long sides of the strip are prescribed.This research was carried out while the first author held a visiting appointment at Cornell University and was partially supported by NSF grant #DMS-9100876.     

Existence of β-martingale solutions of stochastic evolution functional equations of parabolic type with measurable locally bounded coefficients

We prove a theorem on the existence of ??-martingale solutions of stochastic evolution functional equations of parabolic type with Borel measurable locally bounded coefficients. A ??-martingale solution of a stochastic evolution functional equation is understood as a martingale solution of a stochastic evolution functional inclusion constructed on the basis of the equation. We find sufficient conditions for the existence of ??-martingale solutions that do not blow up in finite time.     

Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

This paper deals with the quasilinear degenerate Keller–Segel system (KS) of parabolic–parabolic type. The global existence of weak solutions to (KS) is established when $q<m+\frac{2}{N}$ (m denotes the intensity of diffusion and q denotes the nonlinearity) without restriction on the size of initial data; note that $q=m+\frac{2}{N}$ corresponds to generalized Fujita?s exponent. The result improves both Sugiyama (2007) [14, Theorem 1] and Sugiyama and Kunii (2006) [15, Theorem 1] in which it is assumed that $q?m$.     

On the ε-optimal control of quasilinear integral equations

We consider an optimal control problem for the stochastic quasilinear integro-functional equation , with cost functional . Two controls are introduced: a program control and a feedback control, generated by the optimal control of the problem under consideration when=0. It is proved that both controls are-optimal for the original problem.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 96–100, 1986.     

Extended thermodynamics—a theory of symmetric hyperbolic field equations

Extended thermodynamics is based on a set of equations of balance which are supplemented by local and instantaneous constitutive equations so that the field equations are quasi-linear differential equations of first order. If the constitutive functions are subject to the requirements of the entropy principle, one may write them in symmetric hyperbolic form by a suitable choice of fields. The kinetic theory of gases, or the moment theories based on the Boltzmann equation, provide an explicit example for extended thermodynamics. The theory proves its usefulness and practicality in the successful treatment of light scattering in rarefied gases. It would seem that extended thermodynamics is worthy of the attention of mathematicians. It may offer them a non-trivial field of study concerning hyperbolic equations, if ever they get tired of the Burgers equation. Dedicated to Jürgen Sprekels on the occasion of his 60th birthday     

Finding the coefficient multiplying the time derivative in quasilinear parabolic equations in Hölder spaces

We justify statements in Hölder classes of ill-posed inverse problems with terminal observation for parabolic equations with unknown coefficients multiplying the time derivative. On the basis of the duality principle, we prove sufficient conditions for the uniqueness of solutions in these classes. We present examples in which the uniqueness property is lost if the set of admissible solutions is extended and examples of instability of the solutions with respect to errors in the input data. We justify the quasisolution method for constructing approximate solutions stable in these Hölder classes.     

Long-term analysis of degenerate parabolic equations in ℝ N

Longtime behavior of degenerate equations with the nonlinearity of polynomial growth of arbitrary order on the whole space R^N is considered. By using -trajectories methods, we proved that weak solutions generated by degenerate equations possess an (L_U² (R^N), L_loc² (R^N))-global attractor. Moreover, the upper bounds of the Kolmogorov ε-entropy for such global attractor are also obtained.     

Extension of Euler’s method to parabolic equations

Euler generalized d’Alembert’s solution to a wide class of linear hyperbolic equations with two independent variables. He introduced in 1769 the quantities that were rediscovered by Laplace in 1773 and became known as the Laplace invariants. The present paper is devoted to an extension of Euler’s method to linear parabolic equations with two independent variables. The new method allows one to derive an explicit formula for the general solution of a wide class of parabolic equations. In particular, the general solution of the Black–Scholes equation is obtained.     

Existence and concentration behavior of sign-changing solutions for quasilinear Schr?dinger equations equations

We consider the quasilinear Schrdinger equations of the form-ε~2?u + V(x)u- ε~2?(u2)u = g(u), x ∈ R~N,where ε 0 is a small parameter, the nonlinearity g(u) ∈ C~1(R) is an odd function with subcritical growth and V(x) is a positive Hlder continuous function which is bounded from below, away from zero, and infΛV(x) inf ?ΛV(x) for some open bounded subset Λ of RN. We prove that there is an ε0 0 such that for all ε∈(0, ε0],the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V(x) as ε→ 0~+.