Barenblatt solution and spherically symmetric solutions of the porous media equation

We consider the Cauchy problem of the porous media equation. We show that it is spherically symmetric solution has the same property as Barenblatt solution, with respct to some regularity property.     

Long-time asymptotics for the porous medium equation: The spectrum of the linearized operator

We compute the complete spectrum of the displacement Hessian operator, which is obtained from the confined porous medium equation by linearization around its stationary attractor, the Barenblatt profile. On a formal level, the operator is conjugate to the Hessian of the entropy via similarity transformation. We show that the displacement Hessian can be understood as a self-adjoint operator and find that its spectrum is purely discrete. The knowledge of the complete spectrum and the explicit information about the corresponding eigenfunctions give new insights on the convergence and higher order asymptotics of solutions to the porous medium equation towards its attractor. More precisely, the inspection of the eigenfunctions allows to identify symmetries in R^N${\mathbb{R}}^N$ with flows whose rates of convergence are faster than the uniform, translation-governed bound. The present work complements the analogous study of Denzler & McCann for the fast-diffusion equation.     

A nonlinear oblique derivative boundary value problem for the heat equation Part 2: Singular self-similar solutions

This paper continues the study started in [12]. In the upper half-plane, consider the elliptic equation

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, submitted to the nonlinear oblique derivative boundary condition U_x = UU_z on the axis x = 0. The solution of this problem appears to be the self-similar solution of the heat equation with the same boundary condition. As goes to 0, the function U converges to the non trivial solution U of the corresponding degenerate problem. Moreover there exists z₀ > 0 such that U vanishes for z ≥ z₀, is C^∞ on ]0, z₀[× ₊, is continuous on the boundary x = 0 and is discontinuous on the half-axis {z = 0, x> 0}.     

Structural joining method for the solution of the model Lighthill equation with a regular singular point

Using the structural joining method, we construct a uniformly valid explicit asymptotics of the solution of a perturbed model Lighthill equation with a regular singular point.     

The structure of solutions of a hypergeometric equation system

We consider a system of ordinary differential equations which is a multidimensional analogue of a hypergeometric equation. We study the structure and asymptotics of solutions at the singular points and construct a fundamental system of solutions in a neighborhood of each singular point. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 52–64, January–March, 1999. Translated by V. Mackevičius     

Nonexistence of backward self-similar blowup solutions to a supercritical semilinear heat equation

We consider a Cauchy problem for a semilinear heat equation

with p>p_S where p_S is the Sobolev exponent. If u(x,t)=(T−t)^−1/(p−1)φ((T−t)^−1/2x) for xR^N and t[0,T), where φ is a regular positive solution of

(P)

then u is called a backward self-similar blowup solution. It is immediate that (P) has a trivial positive solution κ≡(p−1)^−1/(p−1) for all p>1. Let p_L be the Lepin exponent. Lepin obtained a radial regular positive solution of (P) except κ for p_S<p<p_L. We show that there exist no radial regular positive solutions of (P) which are spatially inhomogeneous for p>p_L.     

Smooth solution for the porous medium equation in a bounded domain

In this paper, we show the short time existence of the smooth solution for the porous medium equations in a smooth bounded domain:

(0.1)     

A note on the small-time development of the solution to a scalar, non-linear, singular reaction-diffusion equation

In this note, we consider a class of scalar, non-linear, singular (in the sense that the reaction terms in the equation are not Lipschitz continuous) reaction-diffusion equations with positive initial data being of (a) O(x^–) or (b) O(x^–e^{– x}) at large x (dimensionless distance), where , > 0 and are constants. We establish, by developing the small–t (dimensionless time) asymptotic structure of the solution, that the support of the solution becomes finite in infinitesimal time in both cases (a) and (b) above. The asymptotic form for the location of the edge of the support as t 0 is given in both cases.Received: June 6, 2002; revised: May 6 and June 4, 2003     

Global spherically symmetric classical solution to the Navier-Stokes-Maxwell system with large initial data and vacuum

We study the initial boundary value problem to the system of the compressible Navier-Stokes equations coupled with the Maxwell equations through the Lorentz force in a bounded annulus Ω of R3. And a result on the existence and uniqueness of global spherically symmetric classical solutions is obtained. Here the initial data could be large and initial vacuum is allowed.     

Concerning the regularity of the anisotropic porous medium equation

We show that a locally bounded nonnegative weak solution of the anisotropic porous media equation is locally continuous. The proof is based on DiBenedetto's technique called intrinsic scaling; by choosing an appropriate geometry one can deduce energy and logarithmic estimates from which one can implement an iterative method to obtain the regularity result.     

Approximation and solution of a general symmetric functional equation

In this paper, we investigate the Hyers–Ulam–Rassias stability of a general equation f(_φ1(x,y))=_φ2(f(x),f(y))$f\left({\phi }_1(x,y)\right)={\phi }_2(f\left(x\right),f\left(y\right))$ in metric spaces. As a consequence, we obtain some stability results in the sense of Hyers–Ulam–Rassias.     

Mathematical modeling of the dissolving and deposition processes during solution seepage in a porous medium

A mathematical model that describes solution seepage in a porous medium and the processes of mineral dissolving and secondary deposition is proposed. Self-similar solutions describing the motion of the leading and trailing fronts, that is, the boundaries of the complete-dissolving zone, are determined. The main features of the processes under consideration are studied and numerical calculations are performed. It is shown that the model describes well the experimental data on mineral leaching by sulfate solutions. The dynamics of mineral extraction from productive solutions in a medium with a nonuniformacidity distribution are investigated. It is shown that, in the elevated-PH zones, the mineral is dissolved; whereas, in the low-acidity zones, secondary deposition of the mineral occurs. In the latter case, after the work has been completed, the bed may contain more or less considerable mineral resources, depending on the extent of the low-PH zone and its proximity to an extraction well.     

Existence of the self-similar solutions in the heat flow of harmonic maps

It is proved that the heat equations of harmonic maps have self-similar solutions satisfying certain energy condition.     

A very singular solution for the slow diffusion equation with nonlinear convection

We here investigate an existence and uniqueness of the nontrivial, nonnegative solution of a nonlinear ordinary differential equation:

(fm^)″+βrf^′+αf+σ(fq^)′=0     

On the uniqueness of the solution of a nonlinear filtration problem through a porous medium

The purpose of this work is to study a fluid flow through a porous medium governed by a nonlinear Darcy's law. We also impose a nonlinear semi-permeability condition on some part of the boundary of this medium. The main results are the continuity of the free boundary and the uniqueness of the solution. Received May 5, 1996 / In a revised form November 16, 1996 / Accepted December 17, 1996     

Numerical treatment of the spherically symmetric solutions of a generalized Fisher-Kolmogorov-Petrovsky-Piscounov equation

In the present work, the connection of the generalized Fisher-KPP equation to physical and biological fields is noted. Radially symmetric solutions to the generalized Fisher-KPP equation are considered, and analytical results for the positivity and asymptotic stability of solutions to the corresponding time-independent elliptic differential equation are quoted. An energy analysis of the generalized theory is carried out with further physical applications in mind, and a numerical method that consistently approximates the energy of the system and its rate of change is presented. The method is thoroughly tested against analytical and numerical results on the classical Fisher-KPP equation, the Heaviside equation, and the generalized Fisher-KPP equation with logistic nonlinearity and Heaviside initial profile, obtaining as a result that our method is highly stable and accurate, even in the presence of discontinuities. As an application, we establish numerically that, under the presence of suitable initial conditions, there exists a threshold for the relaxation time with the property that solutions to the problems considered are nonnegative if and only if the relaxation time is below a critical value. An analytical prediction is provided for the Heaviside equation, against which we verify the validity of our computational code, and numerical approximations are provided for several generalized Fisher-KPP problems.     

Singular problem for a third-order nonlinear ordinary differential equation arising in fluid dynamics

Results concerning singular Cauchy problems, smooth manifolds, and Lyapunov series are used to correctly state and analyze a singular “initial-boundary” problem for a third-order nonlinear ordinary differential equation defined on the entire real axis. This problem arises in viscous incompressible fluid dynamics and describes self-similar solutions to the boundary layer equation for the stream function with a zero pressure gradient (plane-parallel flow in a mixing layer). The analysis of the problem suggests a simple numerical method for its solution. Numerical results are presented.     

A multi-dimension blow-up problem to a porous medium diffusion equation with special medium void

In this paper, we consider a multi-dimension porous medium equation with special void, a sufficient condition for the solution existing globally and two sufficient conditions for the solution blowing up in finite time are given.