| Introduction |
| Preliminaries / 1： |
| Model Elliptic Problems / I： |
| Classical and weak solutions / 2： |
| Isolated singularities / 4： |
| Pohozaev's identity and nonexistence results / 5： |
| Homogeneous nonlinearities / 6： |
| Minimax methods / 7： |
| Liouville-type results / 8： |
| Positive radial solutions of [delta]u + u[superscript p] = 0 in R[superscript n] / 9： |
| A priori bounds via the method of Hardy-Sobolev inequalities / 10： |
| A priori bounds via bootstrap in L[Characters not reproducible]-spaces / 11： |
| A priori bounds via the rescaling method / 12： |
| A priori bounds via moving planes and Pohozaev's identity / 13： |
| Model Parabolic Problems / II： |
| Well-posedness in Lebesgue spaces / 14： |
| Maximal existence time. Uniform bounds from L[superscript q]-estimates / 16： |
| Blow-up / 17： |
| Fujita-type results / 18： |
| Global existence for the Dirichlet problem / 19： |
| Small data global solutions |
| Structure of global solutions in bounded domains |
| Diffusion eliminating blow-up |
| Global existence for the Cauchy problem / 20： |
| Global solutions with exponential spatial decay |
| Asymptotic profiles for small data solutions |
| Parabolic Liouville-type results / 21： |
| A priori bounds / 22： |
| A priori bounds in the subcritical case |
| Boundedness of global solutions in the supercritical case |
| Global unbounded solutions in the critical case |
| Estimates for nonglobal solutions |
| Blow-up rate / 23： |
| Blow-up set and space profile / 24： |
| Self-similar blow-up behavior / 25： |
| Universal bounds and initial blow-up rates / 26： |
| Complete blow-up / 27： |
| Applications of a priori bounds / 28： |
| A nonuniqueness result |
| Existence of periodic solutions |
| Existence of optimal controls |
| Transition from global existence to blow-up and stationary solutions |
| Decay of the threshold solution of the Cauchy problem |
| Decay and grow-up of threshold solutions in the super-supercritical case / 29： |
| Systems / III： |
| Elliptic systems / 30： |
| A priori bounds by the method of moving planes and Pohozaev-type identities |
| Liouville-type results for the Lane-Emden system |
| A priori bounds by the rescaling method |
| A priori bounds by the L[Characters not reproducible] alternate bootstrap method |
| Parabolic systems coupled by power source terms / 32： |
| Well-posedness and continuation in Lebesgue spaces |
| Blow-up and global existence |
| Blow-up asymptotics |
| The role of diffusion in blow-up / 33： |
| Diffusion preserving global existence |
| Diffusion inducing blow-up |
| Equations with Gradient Terms / IV： |
| Well-posedness and gradient bounds / 34： |
| Perturbations of the model problem: blow-up and global existence / 36： |
| A priori bounds and blow-up rates / 37： |
| Blow-up sets and profiles / 39： |
| Viscous Hamilton-Jacobi equations and gradient blow-up on the boundary / 40： |
| Gradient blow-up and global existence |
| Asymptotic behavior of global solutions |
| Space profile of gradient blow-up |
| Time rate of gradient blow-up |
| An example of interior gradient blow-up / 41： |
| Nonlocal Problems / V： |
| Problems involving space integrals (I) / 42： |
| Blow-up rates, sets and profiles |
| Uniform bounds from L[superscript q]-estimates |
| Universal bounds for global solutions |
| Problems involving space integrals (II) / 44： |
| Transition from single-point to global blow-up |
| A problem with control of mass |
| A problem with variational structure |
| A problem arising in the modeling of Ohmic heating |
| Fujita-type results for problems involving space integrals / 45： |
| A problem with memory term / 46： |
| Appendices |
| Appendix A: Linear elliptic equations / 47： |
| Elliptic regularity |
| L[superscript p]-L[superscript q]-estimates |
| An elliptic operator in a weighted Lebesgue space |
| Appendix B: Linear parabolic equations / 48： |
| Parabolic regularity |
| Heat semigroup, L[superscript p]-L[superscript q]-estimates, decay, gradient estimates |
| Weak and integral solutions |
| Appendix C: Linear theory in L[Characters not reproducible]-spaces and in uniformly local spaces / 49： |
| The Laplace equation in L[Characters not reproducible]-spaces |
| The heat semigroup in L[Characters not reproducible]-spaces |
| Some pointwise boundary estimates for the heat equation |
| Proof of Theorems 49.2, 49.3 and 49.7 |
| The heat equation in uniformly local Lebesgue spaces |
| Appendix D: Poincare, Hardy-Sobolev, and other useful inequalities / 50： |
| Basic inequalities |
| The Poincare inequality |
| Hardy and Hardy-Soboley inequalities |
| Appendix E: Local existence, regularity and stability for semilinear parabolic problems / 51： |
| Analytic semigroups and interpolation spaces |
| Local existence and regularity for regular data |
| Stability of equilibria |
| Self-adjoint generators with compact resolvent |
| Singular initial data |
| Appendix F: Maximum and comparison principles. Zero number / 52： |
| Maximum principles for the Laplace equation |
| Comparison principles for classical and strong solutions |
| Comparison principles via the Stampacchia method |
| Comparison principles via duality arguments |
| Monotonicity of radial solutions |
| Monotonicity of solutions in time |
| Systems and nonlocal problems |
| Zero number |
| Appendix G: Dynamical systems / 53： |
| Appendix H: Methodological notes / 54： |
| Bibliography |
| List of Symbols |
| Index |
| Introduction |
| Preliminaries / 1： |
| Model Elliptic Problems / I： |
図書
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