Global existence of solutions for weakly coupled systems of semi-linear structurally damped σ-evolution models

ABSTRACT

In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped σ-evolution models with different power nonlinearities. By assuming additional $L^m$ regularity on the initial data, with $m\in [1,2)$, we use $(L^m\cap L^2)-L^2$ and $L^2-L^2$ estimates for solutions to the corresponding linear Cauchy problems to prove the global (in time) existence of small data Sobolev solutions to the weakly coupled systems of semi-linear models from suitable function spaces for any $\sigma \ge 1$ and $\delta \in (0,\sigma )$. Moreover, not only the nonexistence of global solution is indicated by using test function method, but also a lifespan estimate is discussed when σ and $\delta \in (0,\sigma )$ are integers.

Keywords:

• Structurally damped σ-evolution equations
• weakly coupled systems
• global existence
• loss of decay
• Harmonic analysis
• critical exponent

2010 Mathematics Subject Classifications:

• 35L30
• 35L56
• 35S05

Acknowledgments

The author would like to express sincere thankfulness to Professor Michael Reissig for his many helpful suggestions and Institute of Applied Analysis for their hospitality. The author wishes to thank the referee for his careful reading of the manuscript and for valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).