Analytic semigroups and reaction-diffusion problems by Lorenzi L., Lunardi A., Metafune G., Pallara D. PDF

By Lorenzi L., Lunardi A., Metafune G., Pallara D.

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15) i,j=1 for some ν > 0. Moreover, if Ω = RN we need that the leading coefficients aij are uniformly continuous. The following results hold. 1 (S. Agmon, [1]) Let p ∈ (1, +∞). (i) Let Ap : W 2,p (RN ) → Lp (RN ) be defined by Ap u = Au. The operator Ap is sectorial in Lp (RN ) and D(Ap ) is dense in Lp (RN ). (ii) Let Ω and A be as above, and let Ap be defined by D(Ap ) = W 2,p (Ω) ∩ W01,p (Ω), Ap u = Au. Then, the operator Ap is sectorial in Lp (Ω), and D(Ap ) is dense in Lp (Ω). 16) i=1 the coefficients bi , i = 1, .

Since both Au and f are continuous in (0, T ], then u is continuous, and u is a classical solution. Now let (b) hold. Since u is continuous at t, then t+h 1 h→0 h u(s)ds = u(t). 13) imply the existence of the limit lim A h→0+ 1 h t+h u(s)ds = u (t) − f (t). t Since A is a closed operator, then u(t) belongs to D(A), and Au(t) = u (t) − f (t). Since both u and f are continuous in (0, T ], then Au is also continuous in (0, T ], so that u is a classical solution. The equivalence of (a ), (b ), (c ) may be proved in the same way.

If b ≤ 0 one gets the same estimate considering γ˜ = {z = x + ix, x ≥ 0}. 12, A is sectorial in X. Let etA be the analytic semigroup generated by A. 7, for Re λ > 0 we have +∞ R(λ, A)f = +∞ e−λt etA f dt = 0 e−λt T (t)f dt 0 hence for every f ∈ X, g ∈ X , +∞ +∞ e−λt etA f, g dt = 0 e−λt T (t)f, g dt. 0 This shows that the Laplace transforms of the scalar-valued functions t → etA f, g , t → T (t)f, g coincide, hence etA f, g = T (t)f, g . Since f, g are arbitrary, etA = T (t). (d) Let us now show that A is an extension of the Laplacian defined in W 2,p (RN ), if X = Lp (RN ), and in Cb2 (RN ) if X = Cb (RN ).

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Analytic semigroups and reaction-diffusion problems by Lorenzi L., Lunardi A., Metafune G., Pallara D.


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