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The Mathematics of Diffusion
Diffusion has been used extensively in many scientific disciplines to model a wide variety of phenomena. The Mathematics of Diffusion focuses on the qualitative properties of solutions to nonlinear elliptic and parabolic equations and systems in connection with domain geometry, various boundary conditions, the mechanism of different diffusion rates, and the interaction between diffusion and spatial heterogeneity. The book systematically explores the interplay between different diffusion rates from the viewpoint of pattern formation, particularly Turing's diffusion-driven instability in both homogeneous and heterogeneous environments, and the roles of random diffusion, directed movements and spatial heterogeneity in the classical Lotka–Volterra competition systems. Interspersed throughout the book are many simple, fundamental and important open problems for readers to investigate.
Degenerate Diffusions
This IMA Volume in Mathematics and its Applications DEGENERATE DIFFUSIONS is based on the proceedings of a workshop which was an integral part of the 1990- 91 IMA program on "Phase Transitions and Free Boundaries". The aim of this workshop was to provide some focus in the study of degenerate diffusion equations, and by involving scientists and engineers as well as mathematicians, to keep this focus firmly linked to concrete problems. We thank Wei-Ming Ni, L.A. Peletier and J.L. Vazquez for organizing the meet ing. We especially thank Wei-Ming Ni for editing the proceedings. We also take this opportunity to thank those agencies whose financial support made the workshop possible: the Army Research Office, the National Science Foun dation, and the Office of Naval Research. A vner Friedman Willard Miller, Jr. PREFACE This volume is the proceedings of the IMA workshop "Degenerate Diffusions" held at the University of Minnesota from May 13 to May 18, 1991.
Nonlinear Diffusion Equations and Their Equilibrium States I
In recent years considerable interest has been focused on nonlinear diffu sion problems, the archetypical equation for these being Ut = D.u + f(u). Here D. denotes the n-dimensional Laplacian, the solution u = u(x, t) is defined over some space-time domain of the form n x [O,T], and f(u) is a given real function whose form is determined by various physical and mathematical applications. These applications have become more varied and widespread as problem after problem has been shown to lead to an equation of this type or to its time-independent counterpart, the elliptic equation of equilibrium D.u + f(u) = o. Particular cases arise, for example, in population genetics, the physics of nu clea...
Nonlinear Diffusion Equations and Their Equilibrium States II
In recent years considerable interest has been focused on nonlinear diffu sion problems, the archetypical equation for these being Ut = ~U + f(u). Here ~ denotes the n-dimensional Laplacian, the solution u = u(x, t) is defined over some space-time domain of the form n x [O,T], and f(u) is a given real function whose form is determined by various physical and mathematical applications. These applications have become more varied and widespread as problem after problem has been shown to lead to an equation of this type or to its time-independent counterpart, the elliptic equation of equilibrium ~u+f(u)=O. Particular cases arise, for example, in population genetics, the physics of nu clear stabi...
Nonlinear Diffusion Equations and Their Equilibrium States II
In recent years considerable interest has been focused on nonlinear diffu sion problems, the archetypical equation for these being Ut = ~U + f(u). Here ~ denotes the n-dimensional Laplacian, the solution u = u(x, t) is defined over some space-time domain of the form n x [O,T], and f(u) is a given real function whose form is determined by various physical and mathematical applications. These applications have become more varied and widespread as problem after problem has been shown to lead to an equation of this type or to its time-independent counterpart, the elliptic equation of equilibrium ~u+f(u)=O. Particular cases arise, for example, in population genetics, the physics of nu clear stabi...
Nonlinear Diffusion Equations and Their Equilibrium States I
In recent years considerable interest has been focused on nonlinear diffu sion problems, the archetypical equation for these being Ut = D.u + f(u). Here D. denotes the n-dimensional Laplacian, the solution u = u(x, t) is defined over some space-time domain of the form n x [O,T], and f(u) is a given real function whose form is determined by various physical and mathematical applications. These applications have become more varied and widespread as problem after problem has been shown to lead to an equation of this type or to its time-independent counterpart, the elliptic equation of equilibrium D.u + f(u) = o. Particular cases arise, for example, in population genetics, the physics of nu clea...
The Lin-Ni's Problem for Mean Convex Domains
The authors prove some refined asymptotic estimates for positive blow-up solutions to $\Delta u+\epsilon u=n(n-2)u^{\frac{n+2}{n-2}}$ on $\Omega$, $\partial_\nu u=0$ on $\partial\Omega$, $\Omega$ being a smooth bounded domain of $\mathbb{R}^n$, $n\geq 3$. In particular, they show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\geq 7$. As a direct consequence, they prove the validity of the Lin-Ni's conjecture in dimension $n=3$ and $n\geq 7$ for mean convex domains and with bounded energy. Recent examples by Wang-Wei-Yan show that the bound on the energy is a necessary condition.
