Proves that for p larger than the Lepin exponent, positive bounded radial entire solutions of u_t = Δu + u^p are steady states, with additional classifications of nonstationary entire and ancient solutions.
On the multiplicity of self-similar solutions of the semilinear heat equation
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
In studies of superlinear parabolic equations \begin{equation*} u_t=\Delta u+u^p,\quad x\in {\mathbb R}^N,\ t>0, \end{equation*} where $p>1$, backward self-similar solutions play an important role. These are solutions of the form $ u(x,t) = (T-t)^{-1/(p-1)}w(y)$, where $y:=x/\sqrt{T-t}$, $T$ is a constant, and $w$ is a solution of the equation $\Delta w-y\cdot\nabla w/2 -w/(p-1)+w^p=0$. We consider (classical) positive radial solutions $w$ of this equation. Denoting by $p_S$, $p_{JL}$, $p_L$ the Sobolev, Joseph-Lundgren, and Lepin exponents, respectively, we show that for $p\in (p_S,p_{JL})$ there are only countably many solutions, and for $p\in (p_{JL},p_L)$ there are only finitely many solutions. This result answers two basic open questions regarding the multiplicity of the solutions.
fields
math.AP 1years
2019 1verdicts
UNVERDICTED 1representative citing papers
citing papers explorer
-
Entire and ancient solutions of a supercritical semilinear heat equation
Proves that for p larger than the Lepin exponent, positive bounded radial entire solutions of u_t = Δu + u^p are steady states, with additional classifications of nonstationary entire and ancient solutions.