The nonlinear heat equation involving highly singular initial values and new blowup and life span results

In this paper we prove local existence of solutions to the nonlinear heat equation ut=Δu+a|u|αu,t∈(0,T),x=(x1,…,xN)∈RN,a=±1,α>0; with initial value u(0)∈Lloc1(RN{0}), anti-symmetric with respect to x1,x2,…,xm and | u(0) | ≤ C(- 1) m∂1∂2⋯ ∂m(| x| - γ) for x1> 0 , … , xm> 0 , where C> 0 is a constant, m∈ { 1 , 2 , … , N} , 0 < γ< N and 0 < α< 2 / (γ+ m). This gives a local existence result with highly singular initial values. As an application, for a= 1 , we establish new blowup criteria for 0 < α≤ 2 / (γ+ m) , including the case m= 0. Moreover, if (N- 4) α< 2 , we prove the existence of initial values u= λf, for which the resulting solution blows up in finite time Tmax(λf) , if λ> 0 is sufficiently small. We also construct blowing up solutions with initial data λnf such that λn[(1α-γ+m2)-1]Tmax(λnf) has different finite limits along different sequences λn→ 0. Our result extends the known “small lambda” blow up results for new values of α and a new class of initial data. © 2018, Orthogonal Publishing and Springer International Publishing AG, part of Springer Nature.