The asymptotic behavior of solutions of some nonlinear initial-boundary value problems of parabolic type

Date
1990
Authors
Deng, Keng
Major Professor
Advisor
Howard A. Levine
Committee Member
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Altmetrics
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Mathematics
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Mathematics
Abstract

In this thesis, three nonlinear initial-boundary value problems of parabolic type are considered. First, blow up to time derivatives of solutions is shown for a problem with singularities. Secondly, global existence and nonexistence of solutions of a fast diffusion equation are established. Finally, stability-instability analysis is given for Burgers' equation under nonlocal boundary conditions;Let [phi](u) ϵ C[superscript]2(-[infinity], A) be nonnegative, increasing, and satisfy lim[limits][subscript]u→ A[superscript]- [phi](u) = [infinity]. For the problem u[subscript]t = [delta] u + [phi](u) in [omega] x (0, T), u = 0 on [partial][omega] x (0, T) and u(x, 0) = u[subscript]0(x), where [omega] is a bounded convex domain in IR[superscript]n, if u quenches (reaches A in finite time), then the quenching points are in a compact subset of [omega], and u[subscript]t blows up there;A study is taken up of solutions of the fast diffusion equation u[subscript]t = (u[superscript]m)[subscript]xx + [epsilon](1 - u)[superscript]-[alpha], 0 0 on (0, 1) x (0, T) with Dirichlet boundary data. The set of stationary solutions is constructed, via which it is proved that solutions with certain initial data may exist globally or quench in finite time;For Burgers' equation u[subscript]t=u[subscript]xx + [epsilon] uu[subscript]x, 0 0, subject to one of four pairs of nonlocal boundary conditions:(UNFORMATTED TABLE OR EQUATION FOLLOWS) (A) u(0,t)=0, u[subscript]x(1,t)=au[superscript]p(1,t)≤ft(ϵt[limits][subscript]sp01u(x, t)dx)[superscript]q-1[over]2[epsilon] u[superscript]2(1,t); or &(B)-u[subscript]x(0,t)=au[superscript]p(0,t)≤ft(ϵt[limits][subscript]sp01 u(x,t)dx)[superscript]q+1[over]2[epsilon] u[superscript]2(0,t), u(1,t)=0; or & (C) u(0,t) = 0, u[subscript]x(1,t)= au[superscript]p(1,t)≤ft(ϵt[limits][subscript]sp01u(x,t)dx)[superscript]q; or & (D)-u[subscript]x(0,t) = au[superscript]p(0,t) ≤ft(ϵt[limits][subscript]sp01u(x,t)dx)[superscript]q, u(1,t) = 0 & with 0 < [epsilon], p, q < [infinity], (TABLE/EQUATION ENDS)solution diagrams for the steady states are given and the stability to instability of each branch is obtained. Blowup in finite time for some solutions is shown.

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