On a class of nonlocal parabolic equations of Kirchhoff type: Nonexistence of global solutions and blow‐up
Uğur Sert, Sergey Shmarev · Mathematical Methods in the Applied Sciences · 2021
We study the homogeneous Dirichlet problem for the degenerate parabolic equation of the Kirchhoff type whereT > 0, ,n ≥ 2, is a smooth bounded domain. The exponentq(x, t) is a measurable function inQTwith values in an interval [q−, q+] ⊂ (1, ∞). The coefficientsa(·),b(·) are continuous functions defined on . It is assumed thata(s) → 0 ora(s) → ∞ass → 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.