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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
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