On a class of similarity solutions of the porous media equation,III

It is shown how existence questions for general multiparameter eigenvalue problems can be treated quite simply using degree theory. The equations to be solved are W_n(λ)x_n = 0 ≠ x_n, n = 1, 2,…, k, where λ ? $\text{R}$^k and each W_n(λ) is a self-adjoint linear operator on a Hilbert space H_n. The W_n, which may be unbounded, depend continuously on λ in a suitable sense. A coercivity condition for large ∥ λ ∥ is used, and is shown to be equivalent, in the “linear” case, to a standard determinantal definiteness condition.     

Qualitative results for solutions of the steady fisher-KPP equation

In this letter, the homogeneous Dirichlet problem involving the N-dimensional Fisher-KPP equation, a reaction-diffusion model which arises in study of population genetics, is investigated for a class of nonlinear polynomial growth laws. Existence and uniqueness conditions for positive (i.e., physically realistic), steady-state solutions on finite domains, or habitats, are noted and stability questions are addressed. Of particular interest are habitats that can be modeled as open balls. For these cases, two relatively recent and powerful theorems from nonlinear analysis are employed to ascertain important qualitative information. Specifically, these solutions are shown to be strictly decreasing and radially symmetric, as well as achieving a stationary maximum at the habitat's center. In addition, the function spaces containing these solutions are determined. Last, the effects of the solution parameters are investigated numerically for the physically relevant cases of N = 2 and 3, the temporal evolution of a particular solution is illustrated, and connections to nuclear reactor science, as well as other fields, are noted.     

Similarity reductions and similarity solutions for the generalized diffusion equation

Herein, the generalized diffusion equation that encompasses the nonlinear diffusion equation with a source term and the Boussinesq equation in hydrology as its particular form and appears in a wide variety of physical and engineering applications has been analyzed via symmetry method that was developed by Steinberg. According to physical situations, in each case, the similarity variables obtained have led us to an ordinary differential equation, and we acquire some new solutions by solving the ODEs. Copyright © 2015 John Wiley & Sons, Ltd     

Smooth zero-contact-angle solutions to a thin-film equation around the steady state

In the simplest case of a linearly degenerate mobility, we view the thin-film equation as a classical free boundary problem. Our focus is on the regularity of solutions and of their free boundary in the “complete wetting” regime, which prescribes zero slope at the free boundary. In order to rule out of the analysis possible changes in the topology of the positivity set, we zoom into the free boundary by looking at perturbations of the stationary solution. Our strategy is based on a priori energy-type estimates which provide “minimal” conditions on the initial datum under which a unique global solution exists. In fact, this solution turns out to be smooth for positive times and to converge to the stationary solution for large times. As a consequence, we obtain smoothness and large-time behavior of the free boundary.     

On the structure of similarity solutions of a differential equation arising from boundary layer problem

We consider an ordinary differential equation with f(0)=a, f^′(0)=1, f^′(∞):=limt→∞f^′(t)=0, where β is a real constant. The given problem may arise from the study of steady free convection flow over a vertical semi-infinite flat plate in a porous medium, or the study of a boundary layer flow over a vertical stretching wall. In this paper, the structure of solutions for the cases of β?−2 is studied. Combining the results of [B. Brighi, T. Sari, Blowing-up coordinates for a similarity boundary layer equation, Discrete Contin. Dyn. Syst. 5 (2005) 929-948; J.-S. Guo, J.-C. Tsai, The structure of solution for a third order differential equation in boundary layer theory, Japan J. Indust. Appl. Math. 22 (2005) 311-351; J.-C. Tsai, Similarity solutions for boundary layer flows with prescribed surface temperature, Appl. Math. Lett. 21 (1) (2008) 67-73], we conclude that the given problem may possess at most two types solutions for β∈R. Moreover, multiple solutions are also verified for various pairs of (a,β).     

Convex solutions of a general similarity boundary layer equation for power-law fluids

This paper is devoted to a general similarity boundary layer equation for power-law fluids, which includes many important similarity boundary layer problems such as the Falker-Skan equation and the magnetohydrodynamic boundary layer equation which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting power-law fluids along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. By a rigorous mathematical analysis, the uniqueness, existence and nonexistence results for convex solutions, normal convex solutions and generalized convex solutions to the general similarity boundary layer equation are established. Also the asymptotic behavior of the normal convex solutions at the infinity are displayed.     

Conservative similarity solutions of the one-dimensional autonomous parabolic equation

Summary conditions are established on the coefficient functions of the one-dimensional autonomous parabolic equation for the existence of conservative solutions in the first quadrant of the plane of the independent variables length and time. Such a solution is characterized by the condition that its integral over the length variable from zero to infinity exist and be independent of time. Those conservative solutions are considered which can be expressed by means of a dimensionless variable and a function of the time variable (similarity). For the case that the similarity relation is linear in the length variable, coefficient functions are specified that lead to conservative solutions.

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Zusammenfassung Es werden Bedingungen aufgestellt über die Koeffizientenfunktionen der eindimensionalen, autonomen, parabolischen Gleichung für die Existenz konservativer Lösungen imersten Quadranten der Ebene der unabhängigen Variablen Länge und Zeit. Eine derartige Lösung ist dadurch charakterisiert, dass ihr Integral über die Längenvariable von Null bis Unendlich existiert und unabhängig von der Zeit ist. Es werden konservative Lösungen betrachtet, die mit Hilfe einer dimensionslosen Variablen ausgedrückt werden könen, die als Quotient einer Funktion der Längenvariablen und einer Funktion der Zeitvariablen definiert ist (Ähnlichkeit). Koeffizientenfunktionen, die zu konservativen Lösungen führen, werden für den Fall angegeben, dass diese Ähnlichkeitsbeziehung linear in der Längenvariablen ist.

     

A note on some similarity solutions of the diffusion equation

Summary A first integral of the diffusion equation exists for similarity solutions when the diffusivity obeys a power or exponential law. Structure of the solutions in both cases and connection to an optimization result are discussed for an arbitrary diffusivity.

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Résumé Une intégrale première de l'équation de diffusion existe pour les solutions similaires quand le coefficient de diffusion obéit une loi de puissance arbitraire ou une loi exponentielle. La structure des solutions est discutée dans les deux cas ainsi que leur relation avec les résultats d'optimisation pour une diffusion arbitraire.

     

On the measurable solutions of a functional equation

The measurable solutions ${f:\mathbb{R}^{3}\setminus\{0\}\to\mathbb{C}\setminus\{0\}\, {\rm and}\, (t,s)\mapsto G(t,s)\in\mathbb{C}\setminus\{0\},\, s\in\mathbb{R}^{3},\, t>|s| >0 }$ of the functional equation $$f(x)f(y)=G\left(|x|+|y|,x+y\right),\quad x,y\in\mathbb{R}^{3}, x\times y\neq 0$$ are considered and it is proved that they are continuous.     

Existence and uniqueness of similarity solutions of imbibition equation

The existence and uniqueness of the similarity profiles for the imbibition phenomenon which may arise due to the difference in the wetting abilities of the two immiscible fluids involved in the displacement process through porous media is discussed. By assuming the validity of the Darcy’s law, a mathematical model has been described and it is found that the investigated flow system is governed by a nonlinear diffusivity type equation. The existence and uniqueness of its similarity solutions have been proved by considering the bounds on the saturation coefficient,N(S _w) which is regarded as positive and piecewise continuously differentiable.     

On the solutions for an Ostrovsky type equation

In this paper, we consider an Ostrovsky type equation that includes the regularized short pulse, the Korteweg–deVries and the modified Korteweg–deVries ones. We prove the well-posedness of the solutions for the Cauchy problem associated with these equations.     

On solutions of a boundary value problem for the biharmonic equation

We study the uniqueness of the solution of a boundary value problem for the biharmonic equation in unbounded domains under the assumption that the generalized solution of this problem has a bounded Dirichlet integral with weight |x|^a. Depending on the value of the parameter a, we prove uniqueness theorems or present exact formulas for the dimension of the solution space of this problem in the exterior of a compact set and in a half-space.     

On positive solutions of a semilinear equation

A semilinear elliptic equation is considered in a domain with smooth boundary. The authors prove the existence and uniqueness of positive solutions of different types, singular at an inner point, subject to the Dirichlet boundary conditions. Bibliography: 13 titles. Translated from Trudy Seminara imeni I., G. Petrovskogo. No. 18, pp. 157–169, 1995.     

On exact solutions of a Sobolev equation

A nonlinear Sobolev-type equation that can be used to describe nonstationary processes in the semiconductor medium is studied. A number of families of exact solutions of this equation that can be expressed in terms of elementary functions and quadratures is obtained; some of these families contain arbitrary sufficiently smooth functions of one argument. The qualitative behavior of the resulting solutions is analyzed.