On a fractional partial differential equation with dominating linear part

It is proved that there is a (weak) solution of the equation u_t=a*u_xx+b*g(u_x)_x+f, on ?⁺ (where * denotes convolution over (?∞, t)) such that u_x is locally bounded. Emphasis is put on having the assumptions on the initial conditions as weak as possible. The kernels a and b are completely monotone and if a(t)=t^?α, b(t)=t^?β, and g(ξ)～sign(ξ)∣ξ∣^γ for large ξ, then the main assumption is that α>(2γ+2)/(3γ+1)β+(2γ?2)/(3γ+1). © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.