The ebook offers with the lifestyles, area of expertise, regularity, and asymptotic habit of recommendations to the preliminary worth challenge (Cauchy challenge) and the initial-Dirichlet challenge for a category of degenerate diffusions modeled at the porous medium sort equation $u_t = \Delta u^m$, $m \geq 0$, $u \geq 0$. Such versions come up in plasma physics, diffusion via porous media, skinny liquid movie dynamics, in addition to in geometric flows resembling the Ricci circulate on surfaces and the Yamabe circulate. The technique offered to those difficulties makes use of neighborhood regularity estimates and Harnack sort inequalities, which yield compactness for households of ideas. the speculation is sort of whole within the gradual diffusion case ($m>1$) and within the supercritical quick diffusion case ($m_c < m < 1$, $m_c=(n-2)_+/n$) whereas many difficulties stay within the variety $m \leq m_c$. All of those facets of the speculation are mentioned within the publication.

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There exists a non-decreasing function H (M) with 0 ≤ H (M) ≤ M/2, such that if Q(R) ⊆ QT , |u| ≤ M in M(R) for 0 < M < C1 , and u(x, t) ≤ M/2 for (x, t) ∈ Q(R/2), then |Q(R) ∩ {u ≤ M − H (M)}| ≥ H (M) |Q(R)| . Proof. Let H (M) = Hˆ (M)/(2C1 + 1), with Hˆ (M) as in the previous lemma. We shall argue by contradiction. Assume that the result were false. Write (M − u) dx dt = Q(R) + A1 where A1 = Q(R) ∩ {u ≤ M − H (M)} A2 = Q(R) ∩ {u > M − H (M)}. 30) A2 46 1 Local regularity and approximation theory Therefore (M − u) dx dt ≤ 2 C1 |Q(R) ∩ {u ≤ M − H (M)}| + H (M) |Q(R)| Q(R) ≤ Hˆ (M)|Q(R)|.

We denote by Dx0 the part which contains x0 and Dz the domain which is obtained by reflecting Dx0 across the plane . Also, let T denote the transformation of reflection across . 2) in the same Dx0 ⊂ QN \ E, while 1 − u(x, cylinder. Moreover 1 − u(T ˜ x, t) ≤ 1 − u(x, ˜ t) 1 − u(z, ˜ t0 ) ≥ on its parabolic boundary. 23). 21) and all the constants depending only on dimension and η. Therefore the lemma follows. 4 via simple rescaling. 5. 2) on the unit cube Q, with 0 ≤ u ≤ 1, then max u − min u =: osc u ≤ 1 − δ.

24) ∇η2 ((w − k)+ )2 . β (M − k − s) s ds 0 so that ∂w ∂ B (w − k)+ = β (M − w) (w − k)+ ∂t ∂t and also B (w − k)+ ≥ (w − k)+ 2 Bˆ (w − k)+ , M − k ˆ Since k ≤ M/2, u < M and w = M − u, it follows that by the definition of B. 6 and the above inequality B (w − k)+ ≥ (w − k)+ 2 μ2 (M − k) ≥ (w − k)+ 2 μ2 (M/2). We also have + B((w − k) ) ≤ (w − k) (w−k)+ + β (M − k − s) ds 0 ≤ (w − k)+ [β(M − k) − β(M − w)] ≤ C (w − k)+ Let I= β (M − w)(w − k)+ ∂w 2 η dxdt. ∂t 2 . 25) . Without loss of generality we can assume μ2 (M/2) ≤ 3/4.