Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l ² + (n ? 2)l = ω ₁. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u ₀ ≥ 0.     

Global solvability for quasilinear parabolic equation with inhomogeneous density and a source

We study the Cauchy problem for quasilinear parabolic equation with inhomogeneous density and a source. We show that this problem has a global solution under the assumptions that initial datum is small enough in the integral sense and the source term has overcritical behaviour. The sharp estimates of a solution is obtained as well.     

Cauchy problem for a quasilinear parabolic equation with a source term and an inhomogeneous density

The following quasilinear parabolic equation with a source term and an inhomogeneous density is considered:

$\rho (x)\frac{{\partial u}}{{\partial t}} = div(u^{m - 1} \left| {Du} \right|^{\lambda - 1} Du) + u^p $

. The conditions on the parameters of the problem are found under which the solution to the Cauchy problem blows up in a finite time. A sharp universal (i.e., independent of the initial function) estimate of the solution near the blowup time is obtained.

     

Critical exponent and critical blow-up for quasilinear parabolic equations

B ₂-groups are special (torsion-free) abelian Butler groups. The interest in this class of groups comes from representation theory. A particular functor, also called Butler functor, connects algebraic properties of the category of free abelian groups with (a few) distinguished subgroups with these Butler groups. This helps to understand Butler groups and caused lots of activities on Butler groups. Butler groups were originally defined for finite rank, however a homological connection discovered by Bican and Salce opened the investigation of Butler groups of infinite rank. Despite the fact that classifications of Butler groups are possible under restriction even for infinite rank (see a forthcoming paper by Files and Göbel [Mathematische Zeitschrift]), general structure theorems are impossible. This is supported by the following very special case of the Main Theorem of this paper, showing that any ring with a free additive group is an endomorphism ring of a Butler group. The result implies the existence of large indecomposable or of large superdecomposable Butler groups as well as the existence of counter-examples for Kaplansky’s test problems.     

Critical extinction exponent for a doubly degenerate non-divergent parabolic equation with a gradient source

We concern with the extinction phenomenon of a non-divergent parabolic equation with a nonlinear gradient source. The critical extinction exponent of the solution is obtained.     

An inverse problem for a quasilinear parabolic equation

Sunto Si considera un problema parabolico sovraderminato in una sola variabile spaziale per l'operatore quasilineare definito da · a₃ o u. Tale operatore contiene un termine incognito a(k {0, 1, 2, 3})del quale si studia la dipendenza dalle condizioni iniziali e alla frontiera. Si determinano due classi, rispettivamente di dati e di soluzioni ammissibili, ed una coppia di metriche rispetto alle quali l'applicazione dati(u, a_k)è lipschitziana. Si mostra infine che tale applicazione si mantiene hölderiana quando le metriche ammesse per i dati sono soltanto del tipo L.

---

---

Lavoro eseguito con contributo del Ministero della Pubblica Istruzione e nell'ambito del Gruppo G.N.A.F.A. del C.N.R.     

Secondary critical exponent,large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values

In this paper, we consider the positive solution of the Cauchy problem for the following doubly degenerate parabolic equation

$$u_t-{\rm div}(|\nabla u|^{p} \nabla u^m)=u^q$$

with p > 0, q > 1,m > 1, and initial value decaying at infinity and give a new secondary critical exponent for the existence of global and nonglobal solutions. Furthermore, the large time behavior and the life spans of solutions are also studied.

     

A critical exponent in a degenerate parabolic equation

We consider positive solutions of the Cauchy problem in $\mathbb{R\,}^n$ for the equation $$u_t=u^p\,\Delta u+u^q,\quad p\geq1,\; q\geq 1$$\nopagenumbers\end and show that concerning global solvability, the number q = p + 1 appears as a critical growth exponent. Copyright © 2002 John Wiley & Sons, Ltd.     

Secondary critical exponent, large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values

In this paper, we consider the positive solution of the Cauchy problem for the following doubly degenerate parabolic equation

Rational approximation and universality for a quasilinear parabolic equation

Approximation theorems, analogous to results known for linear elliptic equations, are obtained for solutions of the heat equation. Via the Cole-Hopf transformation, this gives rise to approximation theorems for one of the simplest examples of a nonlinear partial differential equation, Burgers’ equation.     

The oblique boundary-value problem for a quasilinear parabolic equation

The solvability of the oblique boundary-value problem for quasilinear parabolic nondivergent equations with singularities is studied. Bibliography: 9 titles. To Olga Ladyzhenskaya on the occasion of the jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992. pp. 118–131. Translated by V. I. Ochkur.     

The convergence rate for a semilinear parabolic equation with a critical exponent

We study solutions to the Cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the sense of Joseph and Lundgren and establish the rate of convergence to regular steady states. In the critical case, this rate contains a logarithmic term which does not appear in the supercritical case.     

Boundary value problems for a quasilinear parabolic equation with an unknown coefficient

The work is connected with the mathematical modeling of physical–chemical processes in which inner characteristics of materials are subjected to changes. The considered nonlinear parabolic models consist of a boundary value problem for a quasilinear parabolic equation with an unknown coefficient multiplying the derivative with respect to time and, moreover, involve an additional relationship for a time dependence of this coefficient. For such a system, conditions of unique solvability in a class of smooth functions are studied on the basis of the Rothe method. The proposed approach involves the proof of a priori estimates in the difference-continuous Hölder spaces for the corresponding differential-difference nonlinear system that approximates the original system by the Rothe method. These estimates allow one to establish the existence of the smooth solutions and to obtain the error estimates of the approximate solutions.As examples of applications of the considered nonlinear boundary value problems, the models of destruction of heat-protective composite under the influence of high temperature heating are discussed.