Estimates of blow-up time for a non-local reactive-convective problem modelling ohmic heating of foods

Nikolopoulos, CV; Tzanetis, DE

dc.contributor.author	Nikolopoulos, CV	en
dc.contributor.author	Tzanetis, DE	en
dc.date.accessioned	2014-03-01T01:24:19Z
dc.date.available	2014-03-01T01:24:19Z
dc.date.issued	2006	en
dc.identifier.issn	0013-0915	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/17218
dc.subject	Asymptotic and numerical estimates	en
dc.subject	Blow-up	en
dc.subject	Estimates of blow-up time	en
dc.subject	Non-local hyperbolic equations	en
dc.subject.classification	Mathematics	en
dc.subject.other	THERMISTOR PROBLEM	en
dc.subject.other	EQUATIONS	en
dc.subject.other	CONDUCTIVITY	en
dc.subject.other	TEMPERATURE	en
dc.subject.other	EXISTENCE	en
dc.subject.other	MIXTURES	en
dc.title	Estimates of blow-up time for a non-local reactive-convective problem modelling ohmic heating of foods	en
heal.type	journalArticle	en
heal.identifier.primary	10.1017/S0013091504001610	en
heal.identifier.secondary	http://dx.doi.org/10.1017/S0013091504001610	en
heal.language	English	en
heal.publicationDate	2006	en
heal.abstract	In this work, we estimate the blow-up time for the non-local hyperbolic equation of ohmic type, u(t) + u(x) = lambda f(u)/(integral(1)(0) f(u) dx)(2), together with initial and boundary conditions. It is known that, for f(s), -f'(s) positive and integral(infinity)(0) f(s) ds < infinity, there exists a critical value of the parameter lambda > 0, say lambda, such that for lambda > lambda there is no stationary solution and the solution u(x,t) blows up globally in finite time t, while for lambda <= lambda there exist stationary solutions. Moreover, the solution u(x, t) also blows up for large enough initial data and lambda <= lambda. Thus, estimates for t were found either for lambda greater than the critical value lambda* and fixed initial data u(0)(x) >= 0, or for u(0)(x) greater than the greatest steady-state solution (denoted by w(2) >= w) and fixed lambda <= lambda. The estimates are obtained by comparison, by asymptotic and by numerical methods. Finally, amongst the other results, for given A, A* and 0 < lambda - lambda* << 1, estimates of the following form were found: upper bound epsilon + c(1) ln[c(2)(lambda - lambda)(-1)]; lower bound c(3)(lambda - lambda)(-1/2); asymptotic estimate t* similar to c(4)(lambda - lambda)(-1/2) for f(s) = e(-s). Moreover, for 0 < lambda <= lambda and given initial data u(0)(x) greater than the greatest steady-state solution w(2)(x), we have upper estimates: either c(5) ln(c(6)A(0)(-1) + 1) or epsilon + c(7) ln(c(8)zeta(-1)), where A(0), zeta measure, in some sense, the difference u(0) - w(2) (if u(0) --> w(2)+: then A(0), zeta --> 0+). c(i) > 0 are some constants and 0 < epsilon << 1, 0 < A(0), zeta. Some numerical results are also given.	en
heal.publisher	CAMBRIDGE UNIV PRESS	en
heal.journalName	Proceedings of the Edinburgh Mathematical Society	en
dc.identifier.doi	10.1017/S0013091504001610	en
dc.identifier.isi	ISI:000235872500014	en
dc.identifier.volume	49	en
dc.identifier.issue	1	en
dc.identifier.spage	215	en
dc.identifier.epage	239	en