Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov norms

Abstract:

We consider the Cauchy problem of the nonlinear heat equation u t −Δu=u b ,u(0,x)=u 0 , with b≥2 and b∈N. We prove that initial data u 0 ∈S(R n ) (the Schwartz class) arbitrarily small in the scale invariant Besov-norm B˙ n(b−1)b/2,q−2/b (R n ), can produce solutions that blow up in finite time. The case b=3 answers a question raised by Yves Meyer. Our result also proves that the smallness assumption put in an earlier work by C. Miao, B. Yuan and B. Zhang, for the global-in-time solvability, is essentially optimal.

Año de publicación:

2019

Keywords:

• Besov
• Nonlinear heat equation
• Blowup

Fuente:

scopus