Large Time Asymptotics for Solutions of Nonlinear Partial Differential Equations (eBook)

Preface 5
Contents 6
1 Introduction 8
References 13
2 Large Time Asymptotics for Solutions of Nonlinear First-Order Partial Differential Equations 15
2.1 Introduction 15
2.2 First-order nonlinear partial differential equations -- Someexamples 16
2.3 Decay estimates for solutions of nonlinear first-order partial differential equations 27
2.4 Conclusions 37
References 38
3 Large Time Asymptotic Analysis of Some Nonlinear Parabolic Equations -- Some Constructive Approaches 39
3.1 Introduction 39
3.2 Travelling waves as asymptotics of solutions of initial value problems -- The Burgers equation 40
3.3 Profile at infinity -- Initial boundary value problem for Burgers equation 44
3.4 Travelling waves describing flow in a porous medium 53
3.5 Evolution of a stable profile describing cross-field diffusionin toroidal multiple plasma 62
3.6 Asymptotic solutions describing fast nonlinear diffusion 69
3.7 Large time asymptotic behaviour of periodic solutions of some generalised Burgers equations 80
3.8 Asymptotic behaviour of some generalised Burgers equations via balancing argument 88
3.9 Evolution of travelling waves in generalised Fisher's equations via matched asymptotic expansions 112
3.10 Periodic travelling wave solutions in reaction--diffusion systems 121
3.11 Conclusions 128
References 130
4 Self-similar Solutions as Large Time Asymptotics for Some Nonlinear Parabolic Equations 134
4.1 Introduction 134
4.2 Self-similar solutions as large time asymptotics for linear heat equation 137
4.3 Self-similar solutions as intermediate asymptotics for filtration--absorption model 141
4.4 Analysis of a class of similarity solutions of the porous media equation 150
4.5 Similarity solutions of nonlinear heat conduction problems as large time asymptotics 159
4.6 Large time asymptotics for solutions of the porous mediaequation 166
4.7 Large time behaviour of solutions of a dissipative semilinear heat equation 175
4.8 Large time asymptotics for the solutions of a very fast diffusion equation 181
4.9 Conclusions 189
References 191
5 Asymptotics in Fluid Mechanics 194
5.1 Introduction 194
5.2 Strong explosion in a power law density medium -- Self-similar solutions of first and second kind 195
5.3 Self-similar solutions for collapsing cavities 204
5.4 Large time behaviour of solutions of compressible flow equations with damping 208
5.5 Large time behaviour of solutions of unsteady boundary layer equations for an incompressible fluid 216
5.6 Asymptotic behaviour of velocity profiles in Prandtl boundary layer theory 224
5.7 Conclusions 235
References 235
Index 238