On Non-existence of Global Weak-predictable-random-field Solutions to a Class of SHEs
E. M. Omaba *
Department of Mathematics, Computer Science, Statistics and Informatics, Faculty of Science, Federal University Ndufu-Alike, Ikwo, P.M.B. 1010, Abakaliki, Ebonyi State, Nigeria.
E. Nwaeze
Department of Mathematics, Computer Science, Statistics and Informatics, Faculty of Science, Federal University Ndufu-Alike, Ikwo, P.M.B. 1010, Abakaliki, Ebonyi State, Nigeria.
L. O. Omenyi
Department of Mathematics, Computer Science, Statistics and Informatics, Faculty of Science, Federal University Ndufu-Alike, Ikwo, P.M.B. 1010, Abakaliki, Ebonyi State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
The multiplicative non-linearity term is usually assumed to be globally Lipschitz in most results on SPDEs. This work proves that the solutions fail to exist if the non-linearity term grows faster than linear growth. The global non-existence of the solution occurs for some non-linear conditions on σ. Some precise conditions for existence and uniqueness of the solutions were stated and we have established that the solutions grow in time at most a precise exponential rate at some time interval; and if the solutions satisfy some non-linear conditions then they cease to exist at some finite time t. Our result also compares the non-existence of global solutions for both the compensated and non-compensated Poisson noise equations.
Keywords: Random field solution, weak predictable solution, global solution, Non-existence