A class of nonlocal nonlinear boundary value problems with definite ...
A class of nonlocal nonlinear boundary value problems with definite ...
A class of nonlocal nonlinear boundary value problems with definite ...
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4. Cross-diffusion<br />
The final problem is a limiting equation for the Shigesada-Kawasaki-<br />
Teramoto model <strong>with</strong> cross-diffusion [13]. This problem is the hardest.<br />
Find (v(x), τ) such that τ > 0, and<br />
⎧ Z 1 µ<br />
<br />
τ<br />
τ<br />
a1 − b1 − c1v(x) dx =0,<br />
⎪⎨ 0 v(x) ³ v(x)<br />
τ<br />
´<br />
(C) d2vxx + v · a2 − b2 − c2v =0 in(0, 1),<br />
v<br />
⎪⎩<br />
vx(0) = 0, vx(1) = 0,<br />
v>0 on [0, 1].<br />
where a1,a2,b1,b2,c1,c2,d2 are given positive constants.<br />
We briefly explain the original equation. In 1979, Kawasaki—Shigesada—<br />
Teramoto proposed a cross-diffusion system<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
ut = {(d1 + ρ12v)u} xx + u(a1 − b1u − c1v) (0
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