We denote by the constant which is independent of and the initial data and.
If and the initial data on, thus, it is easy to see that for problem (1.1) simultaneous quenching occurs.
It is considered on a semi-infinite axis and the forcing term and the initial data are assumed compactly supported.
Assume that the heat conductivity κ satisfies (1.4 - 1.6 1.4 - 1.6initianddatheare compatinitialth boundata conditions.
Lemma 4.3 Let s ≥ 2 and the function v ( t, x ) is a solution of problem (3) and the initial data v 0 ( x ) ∈ H s ( R ).
Lemma 4.3 Let s > 3 2 and the function u ( t, x ) is a solution of problem (3) and the initial data u 0 ( x ) ∈ H s ( R ).
Also, since equation (1.1) is not scaling invariant and the initial data are sign-changing, our arguments here are more complicated.
Moreover, there exists a (T_{0}>0), which depends on p, q, m, n, (underline{k}), and the initial data, such that (Tleq T_{0}).
Under suitable assumptions on the relax function g and the initial data, we establish a blow-up result with arbitrary positive initial energy.
Under suitable assumptions on the functions g and the initial data, a blow-up result with arbitrary positive initial energy is established.
Then we prove that when time t → + ∞, the solution decays to zero exponentially under some assumptions on nonlinear functions and the initial data.
