Local Integral Estimates for Quasilinear Equations with Measure Data.
Qiaoyu Tian1, Shengzhi Zhang2, Yonglin Xu1
1School of Mathematics and Computer, Northwest University for Nationalities, Lanzhou, Gansu 730030, China.
Thescientificworldjournal
|June 14, 2016
Summary
This study establishes local integral estimates and nonexistence results for quasilinear and Hessian equations involving nonlinear terms like exp(αu) or u^(p-1). These findings are crucial for understanding the behavior of solutions in these complex mathematical models.
Area of Science:
- Partial Differential Equations
- Nonlinear Analysis
- Mathematical Physics
Background:
- Quasilinear equations and Hessian equations are fundamental in various scientific fields.
- Understanding the behavior of solutions under different nonlinear terms and measures is a key challenge.
- Existing literature often focuses on specific cases, necessitating broader analytical tools.
Purpose of the Study:
- To establish local integral estimates for a class of quasilinear and Hessian equations.
- To derive local nonexistence results for these equations.
- To analyze the impact of nonlinear terms P(u) and measures σ and ω on solution behavior.
Main Methods:
- Utilizing techniques for local integral estimates.
- Developing methods for local nonexistence proofs.
- Analyzing equations of the form -Δpu = σP(u) + ω and Fk[-u] = σP(u) + ω.
- Considering nonlinear terms P(u) ~ exp(αu^(β)) and P(u) = u^(p-1).
Main Results:
- Local integral estimates were successfully established for the studied equations.
- Local nonexistence results were proven, indicating limitations on solution existence.
- The analysis considered a broad class of nonlinearities and measure data.
Conclusions:
- The established estimates and nonexistence results provide a deeper understanding of quasilinear and Hessian equations.
- These findings contribute to the theory of degenerate partial differential equations.
- The results are applicable to problems involving nonlinear diffusion and geometric analysis.
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