Abstract
Solutions of a nonlinear heat equation are numerically computed in the time variable t lying in the complex plane, and possible singularities are sought. It turns out that in the complex half plane \( \{ \mathfrak {R}[t] \ge 0 \}\), where \(\mathfrak {R}\) denotes the real part of a complex number, there is no singularity other than that which exists on the real line. However, if we compute further in the Riemann surface, new singularities are found. A certain nonlinear Schrodinger equation which is associated with our problem is also computed numerically and we propose a conjecture that it is well-posed globally in time.
Highlights
We consider blow-up solutions of a nonlinear heat equation in the following setting of the initial-boundary value problem: ut = uxx + u2 u(0, x) = u0(x) (0 < x < 1), (1) (2)Bunkyo-ku, Tokyo 112-8681, Japan C.-H
As for the initial data, we assume that u0 is continuous, u0(x) ≥ 0 everywhere and u0 ≡ 0. It is well-known that the solution blows up in finite time
Blow-up problems for nonlinear heat equations are studied by many researchers and references are abundant
Summary
We consider blow-up solutions of a nonlinear heat equation in the following setting of the initial-boundary value problem: ut = uxx + u2 u(0, x) = u0(x) (0 < x < 1), (1) (2). With the periodic boundary condition in x. The subscripts t and x imply differentiation. As for the initial data, we assume that u0 is continuous, u0(x) ≥ 0 everywhere and u0 ≡ 0. It is well-known that the solution blows up in finite time. The L∞-norm u(t) L∞ tends to ∞ as t tends to a certain T < ∞. Blow-up problems for nonlinear heat equations are studied by many researchers and references are abundant. See [9,10,25,29,35] for instance
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