2006
DOI: 10.1093/acprof:oso/9780198569039.001.0001
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Abstract: The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. This book provides a presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer, or diffusion.… Show more

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Cited by 512 publications
(598 citation statements)
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References 267 publications
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“…For the existence theory of problem (1.4) we refer to [1,6,7,5,21,22]; in particular we adopt a frame similar to the ones in [1] and [6]. More precisely, we use the following definition.…”
Section: Well Posedness Of the Cauchy Problemmentioning
confidence: 99%
See 1 more Smart Citation
“…For the existence theory of problem (1.4) we refer to [1,6,7,5,21,22]; in particular we adopt a frame similar to the ones in [1] and [6]. More precisely, we use the following definition.…”
Section: Well Posedness Of the Cauchy Problemmentioning
confidence: 99%
“…The fast diffusion, instead, finds a paradigmatic application to the flows in plasma physics. Many results and references can by found in the monographs [2] and [22].…”
Section: Introductionmentioning
confidence: 99%
“…Moreover, defining the positivity set P(t) = {x ∈ Ω : u(t, x) > 0}, one shows that for reasonable initial data, P(t) is expanding with finite speed and that the solution u is regular in the interior of P(t) and has a singularity on its boundary [19,Chap. 14].…”
Section: One Space Dimensionmentioning
confidence: 99%
“…We suppose that D(u) is a non-negative function and note that the equation is degenerate whenever D(u) vanishes. For the convergence analysis of our numerical methods, we will require that D(u) is at least differentiable and D ′ (u) is Lipschitz continuous, while the existence of solutions is guaranteed under the milder assumption of continuity (see [19]). For the eigenvalues clustering results we require that D(u) is also a nondecreasing function.…”
Section: Introductionmentioning
confidence: 99%
“…We have a number of distinct motivations. Firstly, (1.1) has two particularly well-studied special cases, namely the porous medium equation (PME) 1 see Vázquez [19], [20] for accounts of the extensive mathematical theory of these evolution equations. In one dimension there is an obvious relationship between (1.3) and (1.4), with n = m + 1, but -as we shall see -the two can have very different qualitative properties; (1.1) allows the transition between the two be explored.…”
Section: Introductionmentioning
confidence: 99%