On a class of nonlocal parabolic equations of Kirchhoff type: Nonexistence of global solutions and blow‐up.

We study the homogeneous Dirichlet problem for the degenerate parabolic equation of the Kirchhoff type ut−a(‖∇u‖22)Δu=b(‖u‖22)|u|q(x,t)−2uinQT=Ω×(0,T),where T > 0, Ω⊂ℝn, n ≥ 2, is a smooth bounded domain. The exponent q(x, t) is a measurable function in QT with values in an interval [q−, q+] ⊂ (1, ∞). The coefficients a(·), b(·) are continuous functions defined on ℝ+. It is assumed that a(s) → 0 or a(s) → ∞ as s → 0+; therefore, the equation degenerates or becomes singular as ‖ ∇ u(t)‖2 → 0. We prove the local in time solvability of the problem and derive sufficient conditions of the finite time blow‐up of the nonnegative solutions. The upper and lower estimates on the blow‐up moment are found. [ABSTRACT FROM AUTHOR]