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Domain of existence and blowup for the exponential reaction-diffusion equation. (English) Zbl 0928.35080
We investigate the existence, uniqueness, and blowup of solutions to the reaction-diffusion equation $u_t-\Delta u+\lambda e^u, \quad\lambda >0.$ The equation admits in any space dimension $$n>2$$ the singular solution $U(x)=-2\log| x|+\log(2(n-2)/\lambda).$ In dimensions $$n\geq 10$$ this solution plays an important role in defining a domain of existence and uniqueness of solutions of the equation. Thus, the Cauchy problem admits a unique solution for data $$0\leq u_0(x)\leq U(x)$$, while there exists no solution of the equation defined in a strip of the form $$Q=\mathbb{R}^n \times (0,T)$$ for any $$T>0$$ if $$u_0(x)\geq U(x)$$. We prove here that in the physical dimension $$n=3$$ such borderline behaviour fails. Indeed, we show that for every dimension $$4\leq n\leq 9$$ the domain of existence expands in the following precise form: there exists a constant $$c_\# > 0$$, depending on $$n$$, such that the initial data $$u_0(x)=U(x)+c_\#$$ mark the borderline between global existence and instantaneous blowup. In the same dimension range non-uniqueness occurs in a band around the solution $$U(x)$$. The results extend to dimensions $$n=1, 2$$, even if no singular solution like $$U$$ exists.

##### MSC:
 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 35K15 Initial value problems for second-order parabolic equations
##### Keywords:
non-uniqueness; singular solution
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