Nonlinear Parabolic Equations with Regularized Derivatives in Colombeau Algebra
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Nonlinear Parabolic Equations with Regularized Derivatives in Colombeau Algebra Mirjana Stojanovi´c
Received: 5 September 2003 / Accepted: 24 March 2006 / Published online: 16 August 2006 © Springer Science + Business Media B.V. 2006
Abstract We consider several types of nonlinear parabolic equations with singular δ like potential and initial data. To prove the existence-uniqueness theorems we employ regularized derivatives. As a framework we use Colombeau space G p,q ([0, T) ×Rn ), 1 ≤ p, q ≤ ∞, and Colombeau vector space GC1 ,L2 ([0, T), Rn ). AMS Subject Classificassions (2000) 46F30 · 35K55 · 35D05. Key words regularized derivatives · parabolic equations with singularities · Colombeau algebra of generalized functions · Colombeau vector algebras · cut-off method.
1. Introduction For the classical theory of nonlinear parabolic equations cf. [12, 14]. For exhaustive treatment of evolution types of parabolic equations, local and global solutions, link between the singularities and qualitative properties of the solutions cf. [3, 4]. Existence-uniqueness results are established in anisotropic weighted Hölder spaces for local and global solutions. Regularity up to the initial plane of these solutions is achieved. For Colombeau theory of generalized functions cf. [2, 7, 8, 19] and [11]. For nonlinear parabolic equation with Schrödinger operator with δ-potential and singular initial data in Colombeau–Sobolev vector type space GC1 ,W 2 ([0, T), Rn ), cf. [18]. In this paper we consider the Cauchy problem for several types of semilinear parabolic equations in which the nonlinearity is not necessarily Lipschitz-continuous and the initial data are highly singular. Such problems have been treated successfully
M. Stojanovi´c (B) Department of Mathematics and Informatics, University of Novi Sad, Trg D.Obradovi´ca 4, 21 000 Novi Sad, Serbia e-mail: [email protected]
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Acta Appl Math (2006) 92: 1–14
in case of Lipschitz-nonlinearities in Colombeau algebras of generalized functions and regularized spatial derivatives in [21]. Complete treatment of model nonlinear parabolic system with nonlinear conservative term using regularized derivatives with global existence and uniqueness of the solution in the space G L2 (Rn+1 ) when the initial data are in G L2 (Rn ) if forced term g(u) is smooth and globally Lipschitz in u ∈ Rm , g(0) = 0, is given in [21]. It is proved in [21] that weak solutions in (C([0, ∞), L2 (Rn )), of the problem ∂t uε + ∂x f (uε ) ∗ ϕε + g(uε ) = 4uε ∗ ϕ ∗ ϕˇε , uε (0, ·) = u0 (·), is equivalent to the following system of integral equations Z t uε (t) = Tε (t)u0 + Tε (t − τ )( f (uε ) ∗ ∂x ϕε + g(uε (s))ds, 0 ˆ )|2 |ξ |2 t ˆ be the Fouirer inverse of Tˆ ε (t)uˆ = e−|ϕ(εξ where Tε (t) = F (Tˆ ε (t)u) ,where ϕˆ denotes the Fouirer transform of ϕ, (ϕ is a delta sequence), since {Tε (t)}t>0 defines a C0 -semigroup of contractions on L2 (Rn ). This paper is the continuation of the investigation of these results by method combined cut-off and regularization of the nonlinear terms and refined estimates in cor
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