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JournalISSN: 1424-3199

Journal of Evolution Equations 

Springer Science+Business Media
About: Journal of Evolution Equations is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Mathematics & Uniqueness. It has an ISSN identifier of 1424-3199. Over the lifetime, 1057 publications have been published receiving 15352 citations. The journal is also known as: JEE & JEE :.


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Journal ArticleDOI
TL;DR: In this article, it was shown that if S is semi-uniformly stable then the spectrum of A does not intersect the imaginary axis, which is stronger than pointwise stability but strictly weaker than uniform stability.
Abstract: Let S(t) be a bounded strongly continuous semi-group on a Banach space B and – A be its generator. We say that S(t) is semi-uniformly stable when S(t)(A + 1)−1 tends to 0 in operator norm. This notion of asymptotic stability is stronger than pointwise stability, but strictly weaker than uniform stability, and generalizes the known logarithmic, polynomial and exponential stabilities. In this note we show that if S is semi-uniformly stable then the spectrum of A does not intersect the imaginary axis. The converse is already known, but we give an estimate on the rate of decay of S(t)(A + 1)−1, linking the decay to the behaviour of the resolvent of A on the imaginary axis. This generalizes results of Lebeau and Burq (in the case of logarithmic stability) and Liu-Rao and Batkai-Engel-Pruss-Schnaubelt (in the case of polynomial stability).

282 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that for smooth and bounded derivatives, the map f(u) is in W^{s,p} and W^{1,sp} spaces, respectively.
Abstract: Our main result is that, when $f$ is smooth and has bounded derivatives, and when $u$ belongs to the spaces $W^{s,p}$ and $W^{1,sp}$, the map $f(u)$ is in $W^{s,p}$.

212 citations

Journal ArticleDOI
TL;DR: In this paper, an estimate of Carleman type for the one dimensional heat equation was derived for a special pseudo-convex weight function related to the degeneracy rate of a(·).
Abstract: We prove an estimate of Carleman type for the one dimensional heat equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right) \in \left( {0,T} \right) \times \left( {0,1} \right), $$ where a(·) is degenerate at 0. Such an estimate is derived for a special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0, 1] of the semilinear degenerate parabolic equation $$ u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right), $$ where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u.

211 citations

Journal ArticleDOI
TL;DR: In this article, the authors prove the existence of a (C0) contraction semigroup generated by A, the closure of A, on a suitable Lp space, and on C(overline{A} and on \( C(\overline{\Omega}).
Abstract: Let Ω be a bounded subset of RN, \( a \in C^1(\overline\Omega) \) with \( a>0 \) in Ω and A be the operator defined by \( Au := abla\cdot (a abla u) \) with the generalized Wentzell boundary condition.¶¶\( Au + \beta\frac{\partial u}{\partial n} + \gamma u=0\qquad \hbox{on} \quad\partial \Omega. \)¶¶If \( \partial\Omega \) is in C2, β and γ are nonnegative functions in \( C^1(\partial\Omega), \) with β > O, and \( \Gamma:=\{x\in\partial\Omega: a(x)>0\} eq\emptyset \), then we prove the existence of a (C0) contraction semigroup generated by \( \overline{A} \), the closure of A, on a suitable Lp space, \( 1\le p $<$\infty \) and on \( C(\overline{\Omega}).\) Moreover, this semigroup is analytic if \( 1 $<$ p $<$\infty. \)

206 citations

Book ChapterDOI
TL;DR: In this paper, a detailed account of the asymptotic behavior as t → ∞ of the solutions u(x, t) of the PME is presented, where the space dimension can be any integer n ≥ 1.
Abstract: This paper is devoted to present a detailed account of the asymptotic behaviour as t → ∞ of the solutions u(x, t) of the equation $$ {u_{{t = }}}\Delta ({u^{m}}) $$ (0.1) with exponent m > 1, a range in which it is known as the porous medium equation written here PME for short. The study extends the well-known theory of the classical heat equation (HE, the case m = 1) into a nonlinear situation, which needs a whole set of new tools. The space dimension can be any integer n ≥ 1. We will also present the extension of the results to exponents m < 1 (fast-diffusion equation, Fde). For definiteness we consider the Cauchy Problem posed in Q = ℝ n x ℝ+ with initial data $$ u(x,0) = {u_{0}}(x), x \in {\mathbb{R}^{n}} $$ (0.2) chosen in a suitable class of functions. In most of the paper we concentrate on the class X 0 of integrable and nonnegative data, $$ {u_{0}} \in {L^{1}}({\mathbb{R}^{n}}), {u_{0}} \geqslant 0, $$ (0.3) which is natural on physical grounds as the density or concentration of a diffusion process, the height of a ground-water mound, or the temperature of a hot medium (see a comment on the applications at the end). Consequently, we will deal mostly with nonnegative solutions u(x, t) ≥ 0 defined in Q. An existence and uniqueness theory exists for this problem so that for every data uo we can produce an orbit {u(·,t):t > 0} which lives in L 1 (ℝ n ) ∩ L ∞(ℝ n ) and describes the evolution of the process. The solution is not classical for m > 1, but it is proved that there exists a unique weak solution for all m > 0.

183 citations

Performance
Metrics
No. of papers from the Journal in previous years
YearPapers
202326
2022128
2021151
202074
201943
201873