Some blow-up problems for a semilinear parabolic equation with a potential

Some Blow-Up Problems For A Semilinear Parabolic Equation With A Potential

The blow-up rate estimate for the solution to a semilinear parabolic equation ut = ∆u+V (x)|u|p−1u in Ω×(0, T ) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x, 0) = Mφ(x) as M goes to infinity, which have been found in [5], are improved under some reason...

The Blow-up Problem for a Semilinear Parabolic Equation with a Potential

Let Ω be a bounded smooth domain in R . We consider the problem ut = ∆u + V (x)u in Ω × [0, T ), with Dirichlet boundary conditions u = 0 on ∂Ω × [0, T ) and initial datum u(x, 0) = Mu0(x) where M ≥ 0, u0 is positive and compatible with the boundary condition. We give estimates for the blow up time of solutions for large values of M . As a consequence of these estimates we find that, for M larg...

Interior Gradient Blow-up in a Semilinear Parabolic Equation

We present a one dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions we show that gradient blow-up occurs at a single...

Blow-Up in a Fourth-Order Semilinear Parabolic Equation From . . .

The Blow–up Rate for a Semilinear Parabolic Equation with a Nonlinear Boundary Condition

In this paper we obtain the blow-up rate for positive solutions of ut = uxx−λu, in (0, 1)×(0, T ) with boundary conditions ux(1, t) = uq(1, t), ux(0, t) = 0. If p < 2q − 1 or p = 2q − 1, 0 < λ < q, we find that the behaviour of u is given by u(1, t) ∼ (T − t) − 1 2(q−1) and, if λ < 0 and p ≥ 2q − 1, the blow up rate is given by u(1, t) ∼ (T − t) − 1 p−1 . We also characterize the blow-up profil...