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Register a new userA class of non-analytic functions for the global solvability of Kirchhoff equation
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Authors
Fumihiko Hirosawa
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We consider the global solvability to the Cauchy problem of Kirchhoff equation with generalized classes of Manfrins class. Manfrins class is a subclass of Sobolev space, but we shall extend this class as a subclass of the ultradifferentiable class, and we succeed to prove the global solvability of Kirchhoff equation with large data in wider classes from the previous works.
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This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.
We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% begin{equation*} left{ begin{array}{l} -Mleft( int_{mathbb{R}^{3}}leftvert abla urightvert ^{2}dxright) Delta u+mu Vleft( xright) u=Q(x)leftvert urightvert ^{p-2}u+lambda fleft( xright) utext{ in }mathbb{R}^{N}, uin H^{1}left( mathbb{R}^{N}right) ,% end{array}% right. end{equation*}% where $Ngeq 3,2<p<2^{ast }:=frac{2N}{N-2},Mleft( tright) =at+b$ $left( a,b>0right) ,$ the potential $V$ is a nonnegative function in $mathbb{R}% ^{N}$ and the weight function $Qin L^{infty }left( mathbb{R}^{N}right) $ with changes sign in $overline{Omega }:=left{ V=0right} .$ We mainly prove the existence of at least two positive solutions in the cases that $% left( iright) $ $2<p<min left{ 4,2^{ast }right} $ and $0<lambda <% left[ 1-2left[ left( 4-pright) /4right] ^{2/p}right] lambda _{1}left( f_{Omega }right) ;$ $left( iiright) $ $pgeq 4,lambda geq lambda _{1}left( f_{Omega }right) $ and near $lambda _{1}left( f_{Omega }right) $ for $mu >0$ sufficiently large, where $lambda _{1}left( f_{Omega }right) $ is the first eigenvalue of $-Delta $ in $% H_{0}^{1}left( Omega right) $ with weight function $f_{Omega }:=f|_{% overline{Omega }},$ whose corresponding positive principal eigenfunction is denoted by $phi _{1}.$ Furthermore, we also investigated the non-existence and existence of positive solutions if $a,lambda $ belongs to different intervals.
The aim of this paper is to establish the $H^1$ global well-posedness for Kirchhoff systems. The new approach to the construction of solutions is based on the asymptotic integrations for strictly hyperbolic systems with time-dependent coefficients. These integrations play an important role to setting the subsequent fixed point argument. The existence of solutions for less regular data is discussed, and several examples and applications are presented.
For a class of Kirchhoff functional, we first give a complete classification with respect to the exponent $p$ for its $L^2$-normalized critical points, and show that the minimizer of the functional, if exists, is unique up to translations. Secondly, we search for the mountain pass type critical point for the functional on the $L^2$-normalized manifold, and also prove that this type critical point is unique up to translations. Our proof relies only on some simple energy estimates and avoids using the concentration-compactness principles. These conclusions extend some known results in previous papers.
This paper is devoted to proving the almost global solvability of the Cauchy problem for the Kirchhoff equation in the Gevrey space $gamma^s_{eta,L^2}$. Furthermore, similar results are obtained for the initial-boundary value problems in bounded domains and in exterior domains with compact boundary.
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