Impulsive effects on global existence of solutions for degenerate semilinear parabolic equations

摘要

Let q be a nonnegative real number, and λ and σ be positive constants. This article studies the following impulsive problem: for n = 1, 2, 3,…, xq∂∂t−∂2∂x2u = λƒ(u),0 < x <1, (n − 1)T < t ⩽ nT−, u(x,0) = u0(x),0 ⩽ x ⩽ 1, u(x, nT) = σ u(x, nT−),0 ⩽ x ⩽ 1, u(0, t) = 0 = u(1, t), t > 0ƒ. The number λ∗ is called the critical value if the problem has a unique global solution u for λ < λ∗, and the solution blows up in a finite time for λ > λ∗. For σ < 1, existence of a unique λ∗ is established, and a criterion for the solution to decay to zero is studied. For σ > 1, existence of a unique λ∗ and three criteria for the blow-up of the solution in a finite time are given respectively. It is also shown that there exists a unique T∗ such that u exists globally for T > T∗, and u blows up in a finite time for T < T∗.