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Behavior of the solution of a random semilinear heat equation. (English) Zbl 1152.60077

The authors consider the heat equation on \(\mathbb{R}\times \mathbb{R}\) with a quadratic dissipative term and a stationary random source \(\lambda(t)\) at the origin \[ u_{t}+u_{xx}-u^{2}+\lambda(t)\delta_{0}(x)=0. \] The solution of this equation could describe, under suitable rescaling, the density of a system of annihilating Brownian particles. There is a unique positive bounded solution on \(\mathbb{R}\times \mathbb{R}\times\Omega\) and it can be written as \[ u(t,x,\omega)=6(x+a(t,x,\omega))^{-2}. \] The authors prove that if \(\lambda\) is ergodic, a law of large numbers holds : \(\lim_{x\to\infty} a(0,x)={\overline a}\) a.s.; if in addition \(\lambda\) satisfies some mixing conditions a central limit theorem holds : as \(x\to\infty\), \(x(a(0,x)-{\overline a})\) has a limiting normal distribution. As a consequence, this means that, for every \(t\), as \(x\to\infty\), \[ u(t,x)\sim 6x^{-2}-12{\overline a}x^{-3}+12G(t)x^{-4}, \] with \(G(t)\) a Gaussian random variable.

MSC:

60K37 Processes in random environments
35K55 Nonlinear parabolic equations
60G10 Stationary stochastic processes
60J60 Diffusion processes
35R60 PDEs with randomness, stochastic partial differential equations
37A25 Ergodicity, mixing, rates of mixing
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References:

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