Radially Symmetric Solutions of

We investigate solutions of and focus on the regime and . Our advance is to develop a technique to efficiently classify the behavior of solutions on , their maximal positive interval of existence. Our approach is to transform the nonautonomous equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, , such that as and as . Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies for infinitely many values . At the critical value there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime we prove the existence of a family of “super singular” sign changing singular solutions. 1. Introduction In this paper we investigate the behavior of solutions of where , and . Solutions of (1.1) are time-independent solutions of the nonlinear heat equation Equation (1.1) has been widely studied as a canonical model for where is superlinear [1–6]. Our focus is on radially symmetric solutions of (1.1) which have the form , where , and satisfy We distinguish two classes of solutions of (1.4). The first is nonsingular solutions which are bounded at and satisfy , where is finite. The second class consists of singular solutions that are unbounded at . Equation (1.4) has the known positive singular solution Previous Results (i) The positive singular solution has played a central role in analyzing (1.2). For example, when appropriately chosen, similarity solution methods show how as , where is a constant [2, 5, 7]. (ii) Chen and Derrick [8] developed comparison methods to describe the time evolution of solutions of where is superlinear [1–6]. Their approach is to let positive, time independent solutions act as upper and/or lower bounds for initial values of solutions of (1.6). Their comparison technique allows them to prove either global existence or finite time blowup of solutions. (iii) For the case Caffarelli et al. [9] describe the asymptotic behavior of nonnegative solutions of (1.1) that have an isolated singularity at the origin. (iv) Galaktionov [10] studied sign changing singular solutions of (1.4) on the restricted interval . He set and derived an ODE for , . He let , varied , and gave a numerical study of sign changing solutions on . (v) Other studies of nonsingular solutions of (1.4) have used Pohozaev