On parabolic equations with gradient terms

Elbirki, Asma (2016) On parabolic equations with gradient terms. Doctoral thesis (PhD), University of Sussex.

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Abstract

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.

Item Type: Thesis (Doctoral)
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
Depositing User: Library Cataloguing
Date Deposited: 03 Jan 2017 13:29
Last Modified: 03 Jan 2017 13:29
URI: http://sro.sussex.ac.uk/id/eprint/66012

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