POTSDAM -- Clarkson University Assistant Professor of Mathematics Rana Parshad recently published a paper on stochastic heat equations in a mathematics journal.

The paper appears in Proceedings of the American Mathematical Society and addresses longstanding conjecture in the theory of nonlinear stochastic heat equations, Parshad said. Stochastic partial differential equation s are used in diverse applications, such as modeling waves in random media, interacting populations and viscoelastic materials.

Stochastic heat equations are essentially the basic partial differential equation (PDE) governing heat flow, with a forcing term that is random. The forcing term models whether the medium is heated or cooled.

Once there is a PDE with a forcing term such as u^2 or u^3, the central question for mathematicians is whether a solution exists for all time, or blows up or "explodes" in finite time?

Parshad said mathematicians have previously proved that if the power on the source term is more than three-halves, the stochastic heat equation, on a bounded domain, under Dirichlet boundary conditions, will blow up in finite time. For PDEs, Dirichlet boundary conditions specify the values that a solution needs to take on the boundary of the domain.

This paper shows that in the unbounded domain case, finite time blow up occurs if the power is only more than one.

"Our main result is, in the unbounded domain case, if the power on the source term is more than one, then the solution blows up in finite time," he said.

Parshad collaborated on the paper with Assistant Professor Mohammud Foondun at Loughborough University in the United Kingdom. Now that they have proved blow up for this equation in one dimension, Parshad said, their goal is to see if this result can be extended to two or more dimensions.

"The next step is to see if we can numerically investigate this in two dimensions or higher," he said.

Read the full paper, "On Non-Existence of Global Solutions to a Class of Stochastic Heat Equations," at www.ams.org/journals/proc/0000-000-00/S0002-9939-2015-12036-9/S0002-9939-2015-12036-9.pdf.