Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

We study the existence of positive solutions for the nonlinear Schrodinger equation with the fractional LaplacianFurthermore, we analyse the regularity, decay and symmetry properties of these solutions.

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Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

We consider the local and global Cauchy problem for the generalized Korteweg-de Vries equation , with initial data in homogeneous and nonhomogeneous Besov spaces. This allows us to slightly extend ...

On the Cauchy Problem for the Generalized Korteweg-de Vries Equation

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Sharp global well-posedness for KdV and modified KdV on ℝ and

Reviews - Eigenfunction Expansions Associated with Second-Order Differential Equations. By E. C. Titchmarsh Pp. 175. 20s. 1946. (Oxford University)

Contents Introduction 1. Preliminary results 2. m-dissipative operators 3. The Hille-Yosida-Phillips Theorem and applications 4. Nonhomogeneous equations and abstract semilinear problems 5. The heat equation 6. The Klein-Gordon equation 7. The Schrodinger equation 8. Bounds on global solutions 9. The invariance principle and some applications 10. Stability of stationary solutions 11. Comments Bibliography Notation Index

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An Introduction to Semilinear Evolution Equations

We prove in this paper the asymptotic completeness of the family of solitons in the energy space for generalized Korteweg-de Vries equations in the subcritical case (this includes in particular the KdV equation and the modified KdV equation). This result is obtained as a consequence of a rigidity theorem on the flow close to a soliton up to a scaling and a translation, which has its own interest. The proofs use some tools introduced in a previous paper to prove similar results in the case of critical generalized KdV equation.

Asymptotic Stability of Solitons¶for Subcritical Generalized KdV Equations

We consider the generalized Korteweg-de Vries equations
				ut+uxx+upx=0t, x ∈ R
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				in the subcritical and critical cases p = 2, 3, 4 or 5. Let R j ( t, x ) = Qc j ( x - c j t - x j ), where j ∈ {1, . . . , N }, be N soliton solutions of this equation, with corresponding speeds 0 c 1 c 2 c N . In this paper, we construct a solution u ( t ) of the generalized Korteweg-de Vries equation such that
				[inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] This solution behaves asymptotically as t → +∞ as the sum of N solitons without loss of mass by dispersion. This is an exceptional behavior, indeed, being given the parameters { c j }1≤ j ≤ N , { x j }1≤ j ≤ N , we prove uniqueness of such a solution. In the integrable cases p = 2 and 3, such solutions are explicitly known and their properties were extensively studied in the literature (they are called N -soliton solutions). Therefore, the existence result is new only for the nonintegrable cases. The uniqueness result is new for all cases.

Asymptotic n-soliton-like solutions of the subcritical and critical generalized korteweg-de vries equations

We prove in this paper the stability and asymptotic stability in H1 of a decoupled sum of N solitons for the subcritical generalized KdV equations \(\) The proof of the stability result is based on energy arguments and monotonicity of the local L2 norm. Note that the result is new even for p=2 (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [16].

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Stability and Asymptotic Stability for Subcritical gKdV Equations

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Yvan Martel

Papers

An Introduction to Semilinear Evolution Equations

Asymptotic Stability of Solitons¶for Subcritical Generalized KdV Equations

Asymptotic n-soliton-like solutions of the subcritical and critical generalized korteweg-de vries equations

Stability and Asymptotic Stability for Subcritical gKdV Equations

Blow up for $u_t-\Delta u=g(u)$ revisited