Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions. (English) Zbl 1338.60155

The authors consider an equation \(u_t=\frac\nu 2u_{xx}+\rho(u)\,\dot W\) on \(\mathbb R\) with the initial condition \(u(0)=\mu\), where \(W\) is a space-time white noise, \(\rho\) a real Lipschitz function and \(\mu\) a signed measure on \(\mathbb R\) such that \(e^{-ax^2}\in L^1(|\mu|)\) for every \(a>0\). First, it is proved that there exists a unique solution \(\{u(t,x)\}\) and this solution, as a function of \((t,x)\), is continuous in \(L^p(\Omega)\) for every \(p<\infty\). Secondly, the authors prove upper estimates of \(\mathbb E\,u(t,x)u(\tau,y)\) and \(\mathbb E\,|u(t,x)|^p\) for even integers \(p\geq 2\). If \(\rho^2(x)\geq l^2(c^2+x^2)\) for some \(l>0\) and \(c\in\mathbb R\), then also a lower estimate of \(\mathbb E\,u(t,x)u(\tau,y)\) is derived. If, in particular, \(\rho^2(x)=\lambda^2(\varsigma^2+x^2)\), then an explicit formula for \(\mathbb E\,u(t,x)u(\tau,y)\) is found.
In the second part of the paper, the authors prove lower and upper estimates for the growth indices \[ \underline\lambda(p)=\sup\left\{\alpha>0:\limsup_{t\to\infty}\frac 1t\sup_{|x|\geq\alpha t}\log\mathbb E\,|u(t,x)|^p>0\right\}, \]
\[ \overline\lambda(p)=\inf\left\{\alpha>0:\limsup_{t\to\infty}\frac 1t\sup_{|x|\geq\alpha t}\log\mathbb E\,|u(t,x)|^p<0\right\} \] for \(p\geq 2\). The upper estimates require additionally that \(e^{\beta|x|}\in L^1(|\mu|)\) for some \(\beta>0\), whereas the lower estimates require super-linear growth of \(|\rho|\) and non-negativity of the initial condition \(\mu\). Finally, sufficient conditions for \(\underline\lambda(p)=\overline\lambda(p)=\infty\) to hold are derived in terms of the growth of \(|\rho|\) and explicit formulae for \(\underline\lambda(2)\), \(\overline\lambda(2)\) are proved in the case \(\rho^2(x)=\lambda^2(\varsigma^2+x^2)\).


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G60 Random fields
35R60 PDEs with randomness, stochastic partial differential equations


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