By Angelo Favini, Gabriela Marinoschi

The target of those notes is to incorporate in a uniform presentation type numerous issues on the topic of the speculation of degenerate nonlinear diffusion equations, taken care of within the mathematical framework of evolution equations with multivalued *m*-accretive operators in Hilbert areas. the issues challenge nonlinear parabolic equations regarding situations of degeneracy. extra accurately, one case is because of the vanishing of the time by-product coefficient and the opposite is supplied by way of the vanishing of the diffusion coefficient on subsets of confident degree of the domain.

From the mathematical viewpoint the consequences offered in those notes may be regarded as common ends up in the speculation of degenerate nonlinear diffusion equations. even though, this paintings doesn't search to provide an exhaustive learn of degenerate diffusion equations, yet relatively to stress a few rigorous and effective options for imminent a variety of difficulties concerning degenerate nonlinear diffusion equations, equivalent to well-posedness, periodic recommendations, asymptotic behaviour, discretization schemes, coefficient id, and to introduce suitable fixing tools for every of them.

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**Example text**

1 Well-Posedness for the Cauchy Problem with Fast Diﬀusion 19 In the second step we take θ0 ∈ L2 (Ω) = D(Bε ), f ∈ L2 (0, T ; V ). Since W 1,1 ([0, T ]; V ) is dense in L2 (0, T ; V ) and D(Bε ) = V is dense in L (Ω) we can take the sequences (fn )n≥1 ⊂ W 1,1 ([0, T ]; V ) and (θ0n )n≥1 ⊂ D(Bε ) such that 2 fn → f strongly in L2 (0, T ; V ), θ0n → θ0 strongly in L2 (Ω) as n → ∞. e. 58) for any t ∈ [0, T ]. We stress that ε is ﬁxed. 60) 2 dt + K T + 2ε, due to the strong convergence θ0n → θ0 and fn → f as n → ∞.

On Q. Proof. 27) corresponding to the same data f and θ0 . 27) written for y ∗ and y ∗ , multiply the diﬀerence scalarly in V by u(y ∗ − y ∗ )(t), and integrate over (0, t). 108) where A0 ψ = u(y ∗ − y ∗ ). 7) we obtain 1 u(y ∗ − y∗ )(t) 2 ≤ ρ 2 t 0 Ω 2 V t +ρ 0 Ω u(y ∗ − y ∗ )2 dxdτ u(y ∗ − y ∗ )2 dxdτ + 1 (N MK ku )2 2ρ t 0 u(y ∗ − y ∗ )(τ ) 2 V dτ. 2 Therefore, by Gronwall lemma (see [29]), u(y ∗ − y∗ )(t) V ≤ 0 and we deduce that uy ∗ (t) = uy∗ (t) for any t ∈ [0, T ]. e. on the set Qu where u(x) > 0.

1 Well-Posedness for the Cauchy Problem with Fast Diﬀusion 13 The operator Bε : D(Bε ) ⊂ V → V is single-valued, has the domain v uε v ∈ L2 (Ω); βε∗ D(Bε ) := ∈V and is deﬁned by Bε v, ψ := V ,V Ω v uε ∇βε∗ − K0 x, v uε · ∇ψdx, for any ψ ∈ V. 41) v uε In fact we note that Bε v = Aε and v ∈ D(Bε ) is equivalent to uvε ∈ D(Aε ). Also, it is easily seen that D(Bε ) = V. Indeed, if v ∈ D(Bε ) it follows that v ∗ uε ∈ V by the fact that the inverse of βε is Lipschitz, and from here we get 1,∞ that v ∈ V, since uε ∈ W (Ω).