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تسجيل مستخدم جديدThe nonlinear heat equation involving highly singular initial values and new blowup and life span results
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In this paper we prove local existence of solutions to the nonlinear heat equation $u_t = Delta u +a |u|^alpha u, ; tin(0,T),; x=(x_1,,cdots,, x_N)in {mathbb R}^N,; a = pm 1,; alpha>0;$ with initial value $u(0)in L^1_{rm{loc}}left({mathbb R}^Nsetminus{0}right)$, anti-symmetric with respect to $x_1,; x_2,; cdots,; x_m$ and $|u(0)|leq C(-1)^mpartial_{1}partial_{2}cdot cdot cdot partial_{m}(|x|^{-gamma})$ for $x_1>0,; cdots,; x_m>0,$ where $C>0$ is a constant, $min {1,; 2,; cdots,; N},$ $0<gamma<N$ 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 $0<alphaleq 2/(gamma+m)$, 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 $lambda_n^{[({1over alpha}-{gamma+mover 2})^{-1}]}T_{max}(lambda_n f)$ has different finite limits along different sequences $lambda_nto 0$. Our result extends the known small lambda blow up results for new values of $alpha$ and a new class of initial data.
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