ON DIFFUSION INFLUENCED BEHAVIOR OF SOLUTIONS IN A CLASS OF DIFFUSION-REACTION EQUATIONS

• Main Author: SITOMPUL, PARDOMUAN
• Format: Dissertations
• Language: Indonesia
• Subjects: Matematika
• Online Access: https://digilib.itb.ac.id/gdl/view/23659
• Institution: Institut Teknologi Bandung

We study the behavior of the solutions in a class of reaction-diffusion equation on the interval [0; 1]. The problems are equipped with homogeneous Neumann boundary conditions and nonnegative initial conditions. The study is conducted for three types of reactions, namely logistic, sinusoidal, and modified logistic types. In <br /> <br /> particular, the effects of diffusion to the well-posedness of the problem, and energy decay are studied. The energy of the solution is measured as the cumulative of the <br /> <br /> square of the (spatial) gradient of the solution. <br /> <br /> In the logistic diffusion problem, the energy decays for a sufficiently large diffusion coefficient. To investigate energy behavior for small diffusion coefficient however, <br /> <br /> we construct an approximation solution for logistic diffusion equation by using the perturbation method; small diffusion factor is considered as a perturbation term <br /> <br /> to the logistic equation. Furthermore, the approximation obtained is improved by doing recursion. We develop an improved scheme to approximate the logistic diffusion problems. <br /> <br /> Energy decay occurs also for in Sine reaction equation, for sufficiently large diffusion coefficient. In general, Sine reaction equation has a number of equilibria, some are stable and the rest are unstable. Of interest is competition between reaction and diffusion terms in Sine reaction diffusion equation with symmetric initial <br /> <br /> condition with respect to an unstable equilibrium line. In this case, the volume is invariant. Some upper and lower bounds for energy of equilibrium solution with given diffusion coefficient are obtained. <br /> <br /> Modified logistic equation has a semi stable equilibrium at u = 1. The presence of semi stable equilibrium may trigger blow up (in finite time) phenomena of the solution. The strength of diffusion that prevents blow up is investigated for initial volume less than one. Some lower bounds for diffusion coefficient in the classes of <br /> <br /> linear and cosine initial conditions are established, yielding global solution to the modified logistic diffusion problem.