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In this paper we prove local existence of solutions to the nonlinear heat equation \(u_t = \Delta u +a |u|^\alpha u, \; t\in (0,T),\; x=(x_1,\ldots , x_N)\in {\mathbb {R}}^N,\; a = \pm 1,\; \alpha >0;\) with initial value \(u(0)\in L^1_{\mathrm{{loc}}}({\mathbb {R}}^N{\setminus }\{0\}),\) anti-symmetric with respect to \(x_1,\; x_2,\ldots , x_m\) and \(|u(0)|\le C(-1)^m\partial _{1}\partial _{2}\cdots \partial _{m}(|x|^{-\gamma })\) for \(x_1>0,\ldots , x_m>0,\) where \(C>0\) is a constant, \(m\in \{1, 2,\ldots , N\},\) \(0<\gamma <N\) and \(0<\alpha <2/(\gamma +m).\) This gives a local existence result with highly singular initial values. As an application, for \(a=1,\) we establish new blowup criteria for \(0<\alpha \le 2/(\gamma +m),\) including the case \(m=0.\) Moreover, if \((N-4)\alpha <2,\) we prove the existence of initial values \(u_0 = \lambda f,\) for which the resulting solution blows up in finite time \(T_{\max }(\lambda f),\) if \(\lambda >0\) is sufficiently small. We also construct blowing up solutions with initial data \(\lambda _n f\) such that \(\lambda _n^{[({1\over \alpha }-{\gamma +m\over 2})^{-1}]}T_{\max }(\lambda _n f)\) has different finite limits along different sequences \(\lambda _n\rightarrow 0.\) Our result extends the known “small lambda” blow up results for new values of \(\alpha\) and a new class of initial data.

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