Abstract
In this paper we study a class of semilinear degenerate parabolic equations arising in mathematical finance and in the theory of diffusion processes. We show that blow-up of spatial derivatives of smooth solutions in finite time occurs to initial boundary value problems for a class of degenerate parabolic equations. Furthermore, nonexistence of nontrivial global weak solutions to initial value problems is studied by choosing a special test function. Finally, the phenomenon of blow-up is verified by a numerical experiment.
Highlights
In this paper, we consider the equation uxx(z) + u(z)uy(z) – ut(z) = f, in R × R × (, T), ( . )where z = (x, y, t) denotes the point in R
Where z = (x, y, t) denotes the point in R. This equation arises in mathematical finance [ ] and in the physical phenomena such as diffusion and convection of matter
We restrict our consideration to two cases: the initial boundary value problems of ( . ) and the initial value problems of ( . )
Summary
We restrict our consideration to two cases: the initial boundary value problems of ) and the initial value problems of Antonelli and Pascucci [ ] proved that there exists a unique viscosity solution to the initial value problem for
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