What is it about?
This work studies a partial differential equations that model processes where two distinct nonlinear effects interact, the rate at which the quantity u(x,t) (temperature, population density) spreads depends on both its gradient u and its value u
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Why is it important?
- Porous media flows: Heat or contaminant transport in soils whose density can vary with depth. - Non-Newtonian fluids: Spread of a fluid whose viscosity depends on both shear (gradient) and concentration. - Population dynamics: Modelling species whose dispersal rate and local reproduction both vary nonlinearly with population density. - Standard linear theory (e.g., the classical heat equation) or singly nonlinear problems (e.g., porous medium equation or pp Laplacian diffusion alone) do not cover the interplay of two nonlinearities plus variable coefficients. - Non divergent form complicates energy methods and integration by parts, so establishing existence and qualitative behaviour of solutions requires new estimates and function space techniques. - Reliable existence and uniqueness results give confidence that numerical approximations will converge to the true physical behaviour rather than spurious artefacts.
Perspectives
- Long-time behaviour: Do solutions approach steady states or self-similar profiles? - Finite speed vs. infinite speed propagation: Under what parameter regimes does the “front” of u move with finite speed (like in porous media) versus instantly affect the entire domain? - Designing and analysing discretisations (finite elements, finite volumes) that respect the double nonlinearity and variable density. - Proving convergence of schemes and establishing error bounds. - Coupled systems where uu interacts with other fields (e.g., mechanical deformation, electric potential). - Handling domains with rough boundaries or evolving geometries (moving interfaces). - Calibrating the model against experimental data in soil remediation, polymer processing, or tumour growth, using the theory to guide parameter identification and control strategies.
Mr. Makhmud Bobokandov
National University of Uzbekistan
Read the Original
This page is a summary of: Mathematical modeling of double nonlinear problem of reaction diffusion in not divergent form with a source and variable density, Journal of Physics Conference Series, December 2021, Institute of Physics Publishing,
DOI: 10.1088/1742-6596/2131/3/032043.
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