Semigroups of Linear Operators and Applications to Partial Differential Equations pp 230-251 | Cite as

# Applications to Partial Differential Equations—Nonlinear Equations

Chapter

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## Abstract

In this section we consider a simple application of the results of Section 6.1 to the initial value problem for the following nonlinear Schrödinger equation in ∝ where

^{2}$$\left\{ {_{u(x,0) = {u_0}(x)in{\mathbb{R}^2}}^{\frac{1}{i}\frac{{\partial u}}{{\partial t}} - \Delta u + k{{\left| u \right|}^2}u = 0in]0,\infty [x{\mathbb{R}^2}}} \right.$$

(1.1)

*u*is a complex valued function and*k*a real constant. The space in which this problem will be considered is L^{2}(R^{2}). Defining the linear operator*A*_{ 0 }by*D(A*_{ 0 }*) = H*^{ 2 }(R^{2})and*A*_{ 0 }*u*= —*i*&*u*for*u*ϵ*D*(*A*_{0}) the initial value problem (1.1) can be rewritten as$$\left\{{_{u(0) = {u_0}}^{\frac{{du}}{{dt}} + {A_0}u + F(u) = 0fort>0}}\right.$$

(1.2)

where *F(u) = ik\u\*^{ 2 }*u*.

## Keywords

Linear Operator Classical Solution Global Solution Mild Solution Fractional Power
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## Copyright information

© Springer-Verlag New York, Inc. 1983