Theory of Extended Solutions¶for Fast-Diffusion Equations¶in Optimal Classes of Data.¶Radiation from Singularities

Abstract This paper is devoted to constructing a general theory of nonnegative solutions for the equation called “the fast-diffusion equation” in the literature. We consider the Cauchy problem taking initial data in the set ℬ+ of all nonnegative Borel measures, which forces us to work with singular solutions which are not locally bounded, not even locally integrable. A satisfactory theory can be formulated in this generality in the range 1 > m > mc= max {(N− 2)/N,0}, in which the limits of classical solutions are also continuous in ℝN as extended functions with values in ℝ+∪{∞}. We introduce a precise class of extended continuous solutions ℰc and prove (i) that the initial-value problem is well posed in this class, (ii) that every solution u(x,t) in ℰc has an initial trace in ℬ+, and (iii) that the solutions in ℰc are limits of classical solutions.Our results settle the well-posedness of two other related problems. On the one hand, they solve the initial-and-boundary-value problem in ?× (0,∞) in the class of large solutions which take the value u=∞ on the lateral boundary x∈∂?, t>0. Well-posedness is established for this problem for mc < m > 1 when ? is any open subset of ℝN and the restriction of the initial data to ? is any locally finite nonnegative measure in ?. On the other hand, by using the special solutions which have the separate-variables form, our results apply to the elliptic problem Δf=fq posed in any open set ?. For 1 > q > N/(N− 2)+ this problem is well posed in the class of large solutions which tend to infinity on the boundary in a strong sense.As is well known, initial data with such a generality are not allowed for m≧ 1. On the other hand, the present theory fails in several aspects in the subcritical range 0> m≦mc, where the limits of smooth solutions need not be extended-continuously.

[1]  D. Widder.  The heat equation , 1975 .

[2]  Michael G. Crandall,et al.  Regularizing Effects of Homogeneous Evolution Equations , 1980 .

[3]  A. Friedman,et al.  Nonlinear Parabolic Equations Involving Measures as Initial Conditions , 1981 .

[4]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[5]  L. Caffarelli,et al.  The initial trace of a solution of the porous medium equation , 1983 .

[6]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[7]  L. Peletier,et al.  A very singular solution of the heat equation with absorption , 1986 .

[8]  Sigurd Angenent.  Local existence and regularity for a class of degenerate parabolic equations , 1988 .

[9]  J. Vázquez,et al.  A Nonlinear heat equation with singular diffusivity , 1988 .

[10]  C. Kenig,et al.  Nonnegative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders , 1988 .

[11]  M. Bertsch,et al.  Discontinuous “viscosity” solutions of a degenerate parabolic equation , 1990 .

[12]  Positivity properties of viscosity solutions of a degenerate parabolic equation , 1990 .

[13]  M. Bertsch,et al.  Nonuniqueness of solutions of a degenerate parabolic equation , 1992 .

[14]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[15]  J. Vázquez.  Nonexistence of solutions for nonlinear heat equations of fast-diffusion type , 1992 .

[16]  M. Ughi,et al.  The fast diffusion equation with strong absorption: the instantaneous shrinking phenomenon , 1994 .

[17]  L. Caffarelli,et al.  Fully Nonlinear Elliptic Equations , 1995 .

[18]  J. Vázquez,et al.  Non-uniqueness of solutions of nonlinear heat equations of fast diffusion type , 1995 .

[19]  J. Vázquez,et al.  The maximal solution of the logarithmic fast diffusion equation in two space dimensions , 1997, Advances in Differential Equations.

[20]  Laurent Véron,et al.  Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations , 1997 .

[21]  M. Marcus,et al.  The boundary trace of positive solutions of semilinear elliptic equations: The supercritical case , 1998 .

[22]  M. Marcus,et al.  Initial trace of positive solutions of some nonlinear parabolic equations , 1999 .

[23]  Victor A. Galaktionov,et al.  Asymptotics of the Fast-Diffusion Equation with Critical Exponent , 2000, SIAM J. Math. Anal..

[24]  J. Vázquez,et al.  Obstructions to Existence in Fast-Diffusion Equations , 2002 .

[25]  Manuel del Pino,et al.  The influence of domain geometry in boundary blow-up elliptic problems , 2002 .

[26]  J. Vázquez,et al.  The Pressure Equation in the fast diffusion range , 2003 .

[27]  Juan Luis Vázquez,et al.  Darcy's Law and the Theory of Shrinking Solutions of Fast Diffusion Equations , 2003, SIAM J. Math. Anal..