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Öğe Blow up and quenching for a problem with nonlinear boundary conditions(Texas State University - San Marcos, 2015) Ozalp, N.; Selcuk, B.In this article, we study the blow up behavior of the heat equation ut = uxx with ux(0, t) = up(0, t), up(0, t) = ux(a, t). We also study the quenching behavior of the nonlinear parabolic equation vt = vxx +2v2x =(1-v) with vx(0, t) = (1 - v(0, t))-p+2, vx(a, t) = (1 v(a; t))-q+2. In the blow up problem, if u0 is a lower solution then we get the blow up occurs in a finite time at the boundary x = a and using positive steady state we give criteria for blow up and non-blow up. In the quenching problem, we show that the only quenching point is x = a and vt blows up at the quenching time, under certain conditions and using positive steady state we give criteria for quenching and non-quenching. These analysis is based on the equivalence between the blow up and the quenching for these two equations. © 2015 Texas State University - San Marcos.Öğe The quenching behavior of a nonlinear parabolic equation with a singular boundary condition(Hacettepe University, 2015) Ozalp, N.; Selcuk, B.In this paper, we study the quenching behavior of solution of a nonlinear parabolic equation with a singular boundary condition. We prove finite-time quenching for the solution. Further, we show that quenching occurs on the boundary under certain conditions. Furthermore, we show that the time derivative blows up at quenching point. Also, we get a lower solution and an upper bound for quenching time. Finally, we get a quenching rate and lower bounds for quenching time. © 2015, Hacettepe Journal of Mathematics and Statistics. All rights reserved.Öğe The quenching behavior of a semilinear heat equation with a singular boundary outflux(American Mathematical Society, 2014) Selcuk, B.; Ozalp, N.In this paper, we study the quenching behavior of the solution of a semilinear heat equation with a singular boundary outflux. We prove a finite-time quenching for the solution. Further, we show that quenching occurs on the boundary under certain conditions and we show that the time derivative blows up at a quenching point. Finally, we get a quenching rate and a lower bound for the quenching time. © 2014 Brown University.Öğe Quenching behavior of semilinear heat equations with singular boundary conditions(Texas State University - San Marcos, 2015) Selcuk, B.; Ozalp, N.In this article, we study the quenching behavior of solution to the semilinear heat equation (Formula presented) with f(v) = u-r or (1 – v)–r and (Formula presented) For this, we utilize the quenching problem ut=uxx with ux(0,t)=u-p(0,t), ux(a,t)=(1-u(a,t))-q. In the second problem, if u0 is an upper solution (a lower solution) then we show that quenching occurs in a finite time, the only quenching point is x=0 (x=a) and utblows up at quenching time. Further, we obtain a local solution by using positive steady state. In the first problem, we first obtain a local solution by using monotone iterations. Finally, for f(v)=-v-r ((1-v)-r), if v0 is an upper solution (a lower solution) then we show that quenching occurs in a finite time, the only quenching point is x=0 (x=a) and vt blows up at quenching time. © 2015 Texas State University.
