Nonlinear Volterra Equations of Parabolic Type Due to Singular Kernels

Sufficient conditions for the global existence of a strong solution of the equation ut(t, x) = ∫t0k(t − s) σ(ux(s, x))x, ds + f(t, x) are given. The kernel k is nonincreasing and convex with lim inft ↓ 0 √tk(t)>0, and σ is increasing with 0 < inf σ′(p) ≤ sup σ′(p) < ∞. Sufficiently smooth solutions are shown to be unique