By C.V. Pao
In reaction to the transforming into use of response diffusion difficulties in lots of fields, this monograph supplies a scientific therapy of a category of nonlinear parabolic and elliptic differential equations and their functions those difficulties. it truly is a huge reference for mathematicians and engineers, in addition to a realistic textual content for graduate students.
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Extra resources for Nonlinear Parabolic and Elliptic Equations
Example text
Indeed, for a certain set of parameters of u,O Fisher's model possesses multiple positive steady-state solutions. The question of the existence of multiple positive steady-state solutions for this model will be discussed in Chapter 3. 6) where a, b are positive constants and q ~ o. The same model describes the temperature distribution for the case of an adiabatic feedback. In the case of q = 0 this problem is reduced to the Verhulst-Fisher model in population dynamics. 1) with Co = -a, q = q(x) is a nonnegative upper solution.
14) when c(x) and f3o(x) are not both identically zero, it follows that w either is a positive constant or a nonnegative function. This proves the result w ~ o. The positive property w > 0 in 0 follows again from the maximum principle. 15) It is well known that there exist a countable number of real eigenvalues Ao , All A2, ... , such that 0 :::; Ao :::; Al :::; A2 :::; ... , and their corresponding eigenfunctions rPo, rPlI rP2, ... satisfy the orthogonality property 10 rPi(x)rPj(x)dx = 0 when Ai =1= Aj The eigenfunction rPo corresponding to the smallest eigenvalue Ao possesses the same sign in 0 and is either positive or negative in O.
3) the function W(k) - V 2 W(k) = U;