Analysis of a double nonlinear, non-divergent parabolic equation in an inhomogeneous medium

What is it about?

In this article, we delve into the properties of solutions for a double nonlinear parabolic equation with variable density, which is not in divergence form and includes a source. Our main focus is to explore the existence of weak solutions in appropriate function spaces, the positivity of solutions, the asymptotic behaviour of solutions as time progresses to infinity, comparison principles, and maximum principles for solutions. We used the energy method as a foundational tool in our analysis, leveraging its utility in studying the qualitative properties of solutions. Furthermore, asymptotic techniques play a crucial role in understanding the long-term behaviour of solutions, providing insights into the ultimate dynamics of the system. The existence of weak solutions in suitable function spaces is a fundamental aspect of this study, providing a basis for analyzing the equation under consideration. Through the rigorous analysis of weak solutions, positivity, asymptotic behaviour, and comparison principles, we unveil important insights into the nature of solutions to this equation

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Photo by BoliviaInteligente on Unsplash

Photo by BoliviaInteligente on Unsplash

Why is it important?

Global Existence and Blow-Up: Identifies conditions under which solutions globally exist or blow up. Self-Similar Solutions: Constructs Barenblatt-type solutions and evaluates their asymptotics. Energy Methods: Utilizes these for rigorous proofs of solution behaviour. Critical Parameters: Determines Fujita-type critical exponents for existence and non-existence of solutions. The study provides theoretical insights for nonlinear diffusion processes in physics, biology, and image processing. The mathematical framework aids in understanding phenomena like finite propagation speed and long-term asymptotics.

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