scholarly journals The Cauchy problem for the inhomogeneous porous medium equation

2006 ◽  
Vol 1 (2) ◽  
pp. 337-351 ◽  
Author(s):  
Guillermo Reyes ◽  
◽  
Juan-Luis Vázquez ◽  
Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2843
Author(s):  
Ángel García ◽  
Mihaela Negreanu ◽  
Francisco Ureña ◽  
Antonio M. Vargas

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.


Author(s):  
Nicholas D. Alikakos ◽  
Rouben Rostamian

SynopsisWe consider the Cauchy problem for the porous medium equation in one space dimension, with initial data which are locally integrable. We measure the asymptotic behaviour of the initial data near infinity in an integral sense and relate this to the pointwise rate of growth or decay of solution for large time. The emphasis is on a novel comparison method wherein the initial data are rearranged on the ×-axis to form a sequence of Dirac δ-masses. By using the explicit solution in the latter case, we derive upper and lower bounds for the solution to the original problem by comparisons.


2006 ◽  
Vol 31 (4) ◽  
pp. 497-514 ◽  
Author(s):  
Raúl Ferreira ◽  
Arturo de Pablo ◽  
Guillermo Reyes ◽  
Ariel Sánchez

2020 ◽  
Vol 140 (2) ◽  
pp. 395-407 ◽  
Author(s):  
Damião J. Araújo ◽  
Anderson F. Maia ◽  
José Miguel Urbano

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