Inhomogeneous nonlinear high-order evolution equations
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ORIGINAL ARTICLE
Inhomogeneous nonlinear high-order evolution equations Tarek Saanouni1 Received: 2 November 2018 / Accepted: 11 April 2020 Ó Sociedad Matemática Mexicana 2020
Abstract Well-posedness issues of some high-order evolution equations with inhomogeneous exponential non-linearity are investigated in even space dimensions. Keywords Inhomogeneous high-order evolution equations Ground state Moser–Trudinger inequality Well-posedness
Mathematics Subject Classification Primary 35Q55 Secondary 35B30
1 Introduction First, consider the inhomogeneous high-order wave-type problem ( o2t u þ ðDÞn u þ u þ kjxjl f ðuÞ ¼ 0; ðu; ot uÞjt¼0 ¼ ðu0 ; u1 Þ:
ð1Þ
Second, we are interested on the inhomogeneous high-order heat-type problem ot u þ ðDÞn u þ u þ kjxjl f ðuÞ ¼ 0; uð0; :Þ ¼ u0 : Finally, we treat the inhomogeneous high-order Schro¨dinger-type problem iot u ðDÞn u ¼ kjxjl f ðuÞ; uð0Þ ¼ u0 :
ð2Þ
ð3Þ
Here and hereafter u :¼ uðt; xÞ : Rþ R2n ! K, where K ¼ R for the case of (1)
& Tarek Saanouni [email protected]; [email protected] 1
Department of Mathematics, College of Sciences and Arts of Oklat Al Suqoor, Qassim University, Buraydah, Kingdom of Saudi Arabia
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T. Saanouni
or (2) and K ¼ C for the Schro¨dinger case, l n 2 N, Dn :¼RDðDn1 Þ is the n : iterated Laplacian operator and k ¼ 1. Finally, denote, F :¼ 0 f ðtÞ dt.............’’. Kindly check and amend if necessary‘‘–[ In the Schro¨dinger context (3), f is assumed to take the Hamiltonian form f ðxÞ :¼ xgðx2 Þ for some positive realR smooth : function g and f is extended to C by f ðzÞ :¼ jzjz f ðjzjÞ. Finally, denote, F :¼ 0 f ðtÞ dt and the n derivative ( n D2 ; if n 2 2N; n r :¼ n1 rD 2 ; if n 2 1 þ 2N: High-order semi-linear and quasilinear diffusion operators occur in applications in thin film theory, nonlinear diffusion and lubrication theory, flame and wave propagation, and phase transition at critical Lipschitz points and bistable systems [22]. There exist some relevant quantities in the study of the above equations. Indeed, any solution to (1) satisfies (formally) conservation of the energy EðuðtÞÞ :¼ þk
1 1 1 krn uðtÞk2L2 ðR2n Þ þ kot uðtÞk2L2 ðR2n Þ þ kuðtÞk2L2 ðR2n Þ 2Z 2 2 jxjl FðuðtÞÞ dx
R2n
:¼ EðtÞ: Any solution to (2) satisfies (formally) decay of the energy Z 1 1 2 2 n jxjl FðuðtÞÞ dx JðuðtÞÞ :¼ kr uðtÞkL2 ðR2n Þ þ kuðtÞkL2 ðR2n Þ þ k 2 2 R2n Z t ¼ Jð0Þ kot uðsÞk2L2 ðR2n Þ ds
ð4Þ
0
:¼ JðtÞ: Moreover, any solution to (2) satisfies the following evolution of the mass 1 ot kuðtÞk2L2 ðR2n Þ ¼ krn uðtÞk2L2 ðR2n Þ þ kuðtÞk2L2 ðR2n Þ 2 Z þk jxjl uðtÞf ðuðtÞÞ dx:
ð5Þ
R2n
Solutions of (3) satisfy (formally) the conservation of mass and Hamiltonian 1 MðtÞ :¼ MðuðtÞÞ :¼ kuðtÞk2L2 ðR2n Þ ; 2 Z 1 n HðtÞ :¼ HðuðtÞÞ :¼ kr uðtÞk2L2 ðR2n Þ þ k jxjl FðjuðtÞjÞ dx: 2 R2n To use the previous identities with a minimal regularity, it is natural to call energy space the usual Sobolev space H n ðR2n Þ endowed with the complete norm
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Inhomogeneous nonlinear high-order evolution equations
12 k kH n
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