Global structure of self-similar solutions in a semilinear parabolic equation
This paper is concerned with the structure of solutions of the differential equation \[ u''+ \Biggl({n- 1\over r}+{r\over 2}\Biggr) u'+ {1\over p-1} u+|p|^{p- 1}u= 0,\quad r\in (0,\infty),\quad p>1\tag{1} \] which are related to the radial self-similar solutions of \[ v_t= \Delta v+|v|^{p- 1}v= 0,\quad (t,x)\in (0,\infty)\times \mathbb{R}^n \] via the transformation \[ v(x, t)= t^{-1/(p- 1)}u(r),\quad r= t^{-1/2}|x|. \] With the initial conditions \(u(0)= \alpha\geq 0\), \(u'(0)= 0\) it is known that (1) has a unique global \(C^2\) solution \(u(r; \alpha,p)\). Moreover, the limit \(\lim_{r\to\infty} r^{2/(p- 1)}u(r; \alpha,p)\) exists and is finite. If this limit is zero the solution \(u\) is said to decay rapidly, otherwise it decays slowly. The main result of the paper is the following. For \(k= 0,1,2,\dots\) let \(p_k= 1+2/(n+ 2k)\) and let \(p_{-1}= \infty\) for \(1\leq n\leq 2\), \(n/(n-2)\) for \(n> 2\). Then there are positive \(C^1\) functions \(\alpha_k\) defined for \(p\in (p_k, p_{-1}]\) such that \(u(r; \alpha,p)\) is rapidly decaying for \(\alpha= \alpha_k(p)\) and has precisely \(k\) zeros in \((0,\infty)\). Moreover, for fixed \(p\in (p_k, p_{k-1}]\) the sequence \(\alpha_k(p)\) is increasing and unbounded with \(k\). Finally, \(u(r;\alpha, p)\) is slowly decaying with \(k\) positive zeros for \(0< \alpha< \alpha_k(p)\), slowly decaying with \(i\) positive zeros for \(\alpha_i(p)< \alpha< \alpha_{i+1}(p)\), \(i= k,k+1,k+2\dots\;\). A similar result is proved for the Dirichlet initial conditions \(u(0)= 0\), \(u'(0)= \beta\geq 0\) for the one-dimensional problem \(n= 1\); now \(p_k= 1+ 1/(1+k)\). The proofs rely mainly on known results and counting arguments using the Sturm comparison theorem. The final section deals with solutions of (1) on the whole real line. It is shown that for each \(p> p_k= 1+ 2/(k+1)\) there are exactly two solutions of (1) which have precisely \(k\) zeros and decay rapidly at \(\infty\).
scientific article; zbMATH DE number 1449212