By F. E. Relton
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Additional resources for Applied Differential Equations
2 ∀z ≤ −M0 . Other boundedness estimates We now start the typical technique of the weak theories consisting in multiplying the equation by suitable multipliers, integrating in space or in space-time and then performing a number of integrations by parts and other calculus tricks. In our ﬁrst example we take a function p ∈ C 1 (R) such that p (s) ≥ 0 for all s ∈ R, and let j be the primitive of p with j(0) = 0. 12) with integrals in Ω. The reader should check that this calculation applies to the solutions of both Dirichlet and Neumann problems.
Nonlinear diﬀusion. Related equations 7 play a prominent role in Chapters 18 and 19, where the focusing solutions have a key part in settling the regularity issue. 3 Nonlinear diﬀusion. Related equations The PME is but one example of partial diﬀerential equation in the realm of what is called nonlinear diﬀusion. Work in that wide area has frequent overlaps between the diﬀerent models, both in phenomena to be described, results to be proved and techniques to be used. A quite general form of nonlinear diﬀusion equation, as it appears in the specialized literature, is d ∂t H(x, t, u) = ∂xi (Ai (x, t, u, Du)).
Of course, we have left out the developments for parallel equations and models, though their mathematical development has been closely connected to that of the PME, like (i) The combination of nonlinear diﬀusion and reaction or absorption. This is a classical area where a wide literature exists. (ii) The combined models involving nonlinear diﬀusion and convection, like ut = ∆Φ(u) + ∇ · F(u). This has been a very active area of research in recent years. (iii) Gradient ﬂows and p-Laplacian equations, and their relation with the PME in 1D.
Applied Differential Equations by F. E. Relton