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A mathematician from RUDN University has proven that there are no solutions to functional differential inequalities associated with the Kardar-Parisi-Zhang (KPZ)-type equations, nonlinear stochastic partial differential equations that arise when describing surface growth. The obtained conditions for the absence of solutions will help in studies of polymer growth, the theory of neural networks, and chemical reactions. The article was published in Complex Variables and Elliptic Equations.
The main difficulty with nonlinear partial differential equations is that many of them are not solved exactly. For practical purposes, such equations are solved numerically, and the questions of the existence and uniqueness of their solutions become problems over which scientists have been struggling for decades, and sometimes centuries. One of these problems—Navier-Stokes existence and smoothness—was included in the famous list of Millennium Prize problems: The Clay Mathematical Institute in the U.S. offers a prize of $1 million for solving any of these problems.
Equation - Area - Eg - Plane - Space
Any partial differential equation is defined in a certain area, e.g., on a plane or in a sphere, or in space. Usually, it is possible to find a solution to such equations in a small neighborhood of a point, i.e., a local solution. But it may remain unclear whether there is a global solution for the entire area and how to find it.
Another problem of nonlinear partial differential equations is that their solutions can "blow up," that is, suddenly begin to tend to infinity on finite time intervals. If this happens, it means that there is no general solution. And vice versa, if a general solution does not exist, it means that any local solution found must also "blow up" somewhere. Therefore, it is important to look for conditions under which there is no general solution.
Mathematicians - Inequalities
Mathematicians use differential inequalities in their...
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