This thesis is concerned with the study of the important effect of the gradient term in parabolic problems. More precisely, we study the global existence or nonexistence of solutions, and their asymptotic behaviour in finite or infinite time. Particularly when the power of the gradient term can increase to the power function of the solution. This thesis consists of five parts. (i) Steady-State Solutions, (ii) The Blow-up Behaviour of the Positive Solutions, (iii) Parabolic Liouville-Type Theorems and the Universal Estimates, (iv) The Global Existence of the Positive Solutions, (v) Viscous Hamilton-Jacobi Equations (VHJ). Under certain conditions on the exponents of both the function of the solution and the gradient term, the nonexistence of positive stationary solution of parabolic problems with gradient terms are proved in (i). In (ii), we extend some known blow-up results of parabolic problems with perturbation terms, which is not too strong, to problems with stronger perturbation terms. In (iii), the nonexistence of nonnegative, nontrivial bounded solutions for all negative and positive times on the whole space are showed for parabolic problems with a strong perturbation term. Moreover, we study the connections between parabolic Liouville-type theorems and local and global properties of nonnegative classical solutions to parabolic problems with gradient terms. Namely, we use a general method for derivation of universal, pointwise a priori estimates of solutions from Liouville type theorems, which unifies the results of a priori bounds, decay estimates and initial and final blow up rates. Global existence and stability, and unbounded global solutions are shown in (iv) when the perturbation term is stronger. In (v) we show that the speed of divergence of gradient blow up (GBU) of solutions of Dirichlet problem for VHJ, especially the upper GBU rate estimate in n space dimensions is the same as in one space dimension.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:701627 |
| Date | January 2016 |
| Creators | Elbirki, Asma |
| Publisher | University of Sussex |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://sro.sussex.ac.uk/id/eprint/66012/ |
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