Fibering of solutions of Itô stochastic equations of the parabolic type with power-law nonlinearities (Q946106)

The paper is concerned with a one-dimensional SPDE \[ du=a\Delta u^{\sigma+1}\,dt+bu^\gamma\,dw,\qquad u(0,x)=u_0(x),\quad t\in[0,\tau(\omega)),\quad x\in\mathbb R \] where \(\tau\) is a stopping time, \(w\) is a Brownian motion and \(a,b,\sigma,\gamma\) (\(\gamma\neq 1\)) are non-negative parameters. The author is interested in existence and properties of fibering solutions, i.e. that is of the form \(u(t,x)=r(t)v(xr^l(t))\) where \(r\) is a non-negative random process, \(v\) a non-negative deterministic function and \(l\in\mathbb R\). Under various constellations of the parameters \(a,b,\gamma,\sigma\), existence of a fibering solution is proven, terminal behaviour of \(r\) is described, the shape of the function \(v\) is outlined and the corresponding implications for the solution \(u\) are drawn.