Download e-book for iPad: An Exponential Function Approach to Parabolic Equations by Chin-Yuan Lin

By Chin-Yuan Lin

ISBN-10: 9814616389

ISBN-13: 9789814616386

This quantity is on initial-boundary worth difficulties for parabolic partial differential equations of moment order. It rewrites the issues as summary Cauchy difficulties or evolution equations, after which solves them through the means of simple distinction equations. due to this, the quantity assumes much less heritage and offers a simple strategy for readers to understand.

Readership: Mathematical graduate scholars and researchers within the quarter of research and Differential Equations. it's also reliable for engineering graduate scholars and researchers who're attracted to parabolic partial differential equations.

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Extra resources for An Exponential Function Approach to Parabolic Equations

Example text

Proof. 2, so no proof is needed for it. 2. The material of this section is taken from our article [21]. 6. 2. This will be seen from the following proofs. The nonlinear homogeneous case will be proved first, because, once this is done, the remaining cases become clearer. 1. The Nonlinear Homogeneous Case. We shall use the notions of difference equations introduced in Section 5. For a doubly indexed sequence {ρm,n } = {ρm,n }∞ m,n=0 ∈ S × S of real numbers, define E1 {ρm,n } = {ρm+1,n }; E2 {ρm,n } = {ρm,n+1 }.

PROOF OF THE MAIN RESULTS main4 29 Step 4. Claim that u(t) is the uniform limit of un (t) on [0, T ], where T > 0 is arbitrary. For each t ∈ [0, T ), we have t ∈ [ti−1 , ti ) for some i, so i − 1 = [ νt ]. Thus it follows from Step 3 that, for each t ∈ [0, T ), ui−1 = (I − νA)−(i−1) u0 t = (I − νA)−[ ν ] u0 converges, as ν = Tn −→ 0. 2). Hence, the Ascoli-Arzela theorem [33] will prove that u(t) is the uniform limit of un (t) on [0, T ], as n −→ ∞. 2) gives un (t) − un (τ ) ≤ K v0 |t − τ | for t, τ ∈ [0, T ], so un (t) is equi-continuous in C([0, T ]; X), the real Banach space of continuous functions from [0, T ] to X.

0 Here [a] for each a ∈ R is the greatest integer that is less than or equal to a. Step 3. (The continuity and Lipschitz continuity of U (t)x, [6, page 272], [30, page 136]) For x ∈ D(A), let xl ∈ D(A) be such that xl −→ x as l −→ ∞. Then as in the Case 2 above, we have, for μ ≤ λ < λ0 , Jμn x − Jλm x ≤ (1 − μω)−n x − xl + Jμn xl − Jλm xl + (1 − λω)−m xl − x . 2 and letting μ, λ −→ ∞, where t, τ ≥ 0, it follows that U (t)x − U (τ )x ≤ (e ωt + eωτ ) xl − x + |t − τ |[2eωt + eω(t+τ ) ]|Axl | for all l.

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An Exponential Function Approach to Parabolic Equations by Chin-Yuan Lin


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