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Lower and upper bounds for the explosion times of a system of semilinear SPDEs

S. Sankar, Manil T. Mohan, S. Karthikeyan

Abstract

In this paper, we obtain lower and upper bounds for the blow-up times to a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of explosion times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We also provide an estimate for the probability of the finite-time blow-up. With a suitable choice of parameters, the impact of the noise on the solution is investigated. The above-obtained results are also extended for semilinear SPDEs forced by two dimensional Brownian motions.

Lower and upper bounds for the explosion times of a system of semilinear SPDEs

Abstract

In this paper, we obtain lower and upper bounds for the blow-up times to a system of semilinear stochastic partial differential equations. Under suitable assumptions, lower and upper bounds of explosion times are obtained by using explicit solutions of an associated system of random partial differential equations and a formula due to Yor. We also provide an estimate for the probability of the finite-time blow-up. With a suitable choice of parameters, the impact of the noise on the solution is investigated. The above-obtained results are also extended for semilinear SPDEs forced by two dimensional Brownian motions.
Paper Structure (9 sections, 12 theorems, 184 equations)

This paper contains 9 sections, 12 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. A System of Random PDEs
  3. A Lower Bound for $\tau$
  4. Upper Bound
  5. A More General Case
  6. Probability of blow-up
  7. Semilinear SPDEs with two-dimensional Brownian motions
  8. Probability of finite time blow-up
  9. Conclusion

Key Result

Theorem 3.1

Assume that the conditions (a4) hold, and let the initial values be of the form for some positive constants $L_{1}$ and $L_{2}$ with $L_{1}\leq L_{2}$. Let $\tau_{\ast}$ be given by where $\|\psi\|_{\infty}:=\sup\limits_{x\in D}\psi(x)$. Then $\tau_{\ast}\leq\tau$.

Theorems & Definitions (23)

  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 4.1
  • Remark 4.1
  • proof : Proof of Theorem \ref{['thm4.1']}
  • proof
  • Corollary 5.1
  • Theorem 5.1
  • Corollary 5.2
  • ...and 13 more