Appendix I: The Proof of Proposition 5.4

This section gives the proof of Proposition 5.4. To begin with, the following claims are obvious and will be useful.

Claim 1 If z = 0, u6= u^∗, we have

z⁰ = `(w)q(u) has the same sign with [w· (u − u^∗)]. (5.61) Claim 2 If z = 0, u = u_∗, we have z⁰= 0 and

z⁰⁰= `(w)q⁰(u)v has the same sign with − v. (5.62) Claim 3 If v = 0, g(w)− p(u) 6= 0, u 6= 0, we have

dv⁰= h(u)[g(w)− p(u)] has the same sign with [g(w) − p(u)]. (5.63) Claim 4 If v = 0, g(w)− p(u) = 0, u 6= 0, we have v⁰= 0

dv⁰⁰= h(u)g⁰(w)z has the same sign with z. (5.64) Claim 5 If v6= 0, g(w) − p(u) = 0, we have

[g(w)− p(u)]⁰ = g⁰(w)z− p⁰(u)v > 0,if z≥ 0, v ≥ 0, (z, v) 6= (0, 0). (5.65) [g(w)− p(u)]⁰ = g⁰(w)z− p⁰(u)v < 0,if z≤ 0, v ≤ 0, (z, v) 6= (0, 0). (5.66)

The exit set W⁻must be a subset of ∂W . To find W⁻, we need to analyze the dynamics of (5.31) on each portion of ∂W .

In the portion ∂R\ P . Set ∂R = R⁰∪ R¹∪ R²∪ R³ with

R₀={(u, v, w, z) : u = 0, g(w) − p(u) ≤ 0, v ≤ 0}, R1={(u, v, w, z) : u = u^∗, g(w)− p(u) ≤ 0, v ≤ 0}, R₂={(u, v, w, z) : u < u^∗, g(w)− p(u) = 0, v ≤ 0}, R3={(u, v, w, z) : u < u^∗, g(w)− p(u) < 0, v = 0}.

Now we investigate the behavior of solutions on each R_i. On region R0, we consider the following two subsets.

(1) Assume u = 0, v < 0.

Since v < 0, we know that u⁺< 0 and this implies this set belongs to W⁻. (2) Assume u = 0, v = 0.

In this case, (0, 0, w, z) will stay at W for any (w, z)∈ R². Thus, (0, 0, w, z)∈ J². On region R1, we consider the following four subsets.

(1) Assume u = u_∗, w = w_∗, v < 0.

If z≤ 0, then

[g(w)− p(u)]⁰ = g⁰(w)z− p⁰(u)v≤ −p⁰(u)v < 0.

Hence, the any trajectory of solutions will enter the region R. On the other hand, if z > 0, then w⁺> w∗ and this implies that any trajectory of solutions will enter the region P .

(2) Assume u = u_∗, w < w_∗, v < 0.

In this case, it’s easy to see that any trajectory of solutions will enter region R.

(3) Assume u = u_∗, w = w_∗, v = 0.

If z = 0, it’s obviously that (u∗, 0, w∗, 0) /∈ W⁻. If z > 0, then w⁺> w∗, v⁰ = 0,

[g(w)− p(u)]⁰= [g⁰(w)z− p⁰(u)v] = g⁰(w)z > 0,

dv⁰⁰= cv⁰+ h⁰(u)v[g(w)− p(u)] + h(u)[g(w) − p(u)]⁰ = h(u)[g(w)− p(u)]⁰> 0.

Thus, v⁺> 0, u⁰ > 0 and u⁺> u∗. Then, any trajectory of solutions will enter the region S.

If z < 0, similar to case of z > 0, we can obtain w⁺ < w∗ and u⁺ < u∗. Hence, any trajectory of solutions will enter the region R.

(4) Assume u = u_∗, w < w_∗, v = 0.

In this case, we have

dv⁰= cv + h(u_∗)[g(w)− p(u^∗)] < cv + h(u_∗)[g(w_∗)− p(u^∗)] < 0

and this implies v⁺ < 0. Since u⁰ = v < 0, we then obtain u⁺ < u_∗ and g(w)− p(u^∗) <

g(w∗)− p(u^∗) = 0. Hence, any trajectory of solutions will enter the region R.

On region R₂, we consider the following two subsets.

(1) Assume 0 < u < u∗, g(w)− p(u) = 0, v < 0.

will enter the region P . If z = 0, we have

z⁰ = cz + l(w)q(u) > 0 + l(w)q(u∗) = 0

and this implies z⁺> 0 and w⁺> w_∗. Then any trajectory of solutions will enter the region P . If z < 0, it is easy to check that

[g(w)− p(u)]⁰= g⁰(w)z− p⁰(u)v < g⁰(w)z < 0.

Hence [g(w)− p(u)]⁺< 0 and any trajectory of solutions will enter the region R.

(2) Assume 0 < u < u_∗, g(w)− p(u) = 0, v = 0.

If z≥ 0, by the same arguments as (1), any trajectory of solutions will enter the region P . If z < 0, we have

[g(w)− p(u)]⁰= g⁰(w)z < 0⇒ [g(w) − p(u)]⁺< 0, dv⁰= cv + g(w)− p(u) = 0,

dv⁰⁰= cv⁰+ h⁰(u)v[g(w)− p(u)] + h(u)[g⁰(w)z− p⁰(u)v]

= 0 + 0 + h(u)g⁰(w)z + 0 < 0,

and this implies v⁺< 0. Hence, any trajectory of solutions will enter the region R.

On region R3, we have 0 < u < u_∗, g(w)− p(u) < 0, v = 0. Thus,

dv⁰ = cv + h(u)[g(w)− p(u)] = 0 + h(u)[g(w) − p(u)] < 0.

Hence, any trajectory of solutions will enter the region R. The proof is complete.

In the portion ∂S\ Q, we set ∂S = S¹∪ S²∪ S³ with

S₁={(u, v, w, z)|u = u^∗, g(w)− p(u) ≥ 0, v ≥ 0}, S2={(u, v, w, z)|u > u^∗, g(w)− p(u) = 0, v ≥ 0}, S3={(u, v, w, z)|u > u^∗, g(w)− p(u) > 0, v = 0}.

Now we investigate the behavior of solutions on each S_i. On region S1, we consider the following four subsets.

(1) Assume u = u_∗, w = w_∗, v > 0.

If z≥ 0, then

[g(w)− p(u)]⁰ = g⁰(w)z− p⁰(u)v≥ −p⁰(u)v > 0.

Hence, the any trajectory of solutions will enter the region S.

On the other hand, if z < 0, then w⁺< w∗ and this implies that any trajectory of solutions will enter the region Q.

(2) Assume u = u_∗, w > w_∗, v > 0.

In this case, it’s easy to see that any trajectory of solutions will enter region S.

(3) Assume u = u_∗, w = w_∗, v = 0.

If z = 0, it’s obviously that (u∗, 0, w∗, 0) /∈ W⁻. If z > 0, then w⁺> w∗, v⁰ = 0,

[g(w)− p(u)]⁰= [g⁰(w)z− p⁰(u)v] = g⁰(w)z > 0,

dv⁰⁰= cv⁰+ h⁰(u)v[g(w)− p(u)] + h(u)[g(w) − p(u)]⁰ = h(u)[g(w)− p(u)]⁰> 0.

Thus, v⁺> 0, u⁰ > 0 and u⁺> u∗. Then, any trajectory of solutions will enter the region S.

If z < 0, similar to case of z > 0, we can obtain w⁺ < w∗ and u⁺ < u∗. Hence, any trajectory of solutions will enter the region R.

(4) Assume u = u_∗, w > w_∗, v = 0.

In this case, we have

dv⁰= cv + h(u_∗)[g(w)− p(u^∗)] > cv + h(u_∗)[g(w_∗)− p(u^∗)] = 0

and this implies v⁺ > 0. Since u⁰ = v < 0, we then obtain u⁺ > u_∗ and g(w)− p(u^∗) >

g(w_∗)− p(u^∗) = 0. Hence, any trajectory of solutions will enter the region S.

On region S₂, we consider the following two subsets.

(1) Assume u > u∗, g(w)− p(u) = 0, v > 0.

If z < 0, then u > u∗and g(w)−p(u) = 0 imply that w < w^∗. Hence, any trajectory of solutions will enter the region Q.

If z = 0, we have

z⁰ = cz + l(w)q(u) < 0 + l(w)q(u_∗) = 0

and this implies z⁺< 0 and w⁺< w_∗. Then any trajectory of solutions will enter the region Q.

If z > 0, it is easy to check that

[g(w)− p(u)]⁰= g⁰(w)z− p⁰(u)v > g⁰(w)z > 0.

(2) Assume u > u_∗, g(w)− p(u) = 0, v = 0.

If z≤ 0, by the same arguments as (1), any trajectory of solutions will enter the region Q.

If z > 0, we have

[g(w)− p(u)]⁰= g⁰(w)z > 0⇒ [g(w) − p(u)]⁺> 0, dv⁰= cv + g(w)− p(u) = 0,

dv⁰⁰= cv⁰+ h⁰(u)v[g(w)− p(u)] + h(u)[g⁰(w)z− p⁰(u)v]

= 0 + 0 + h(u)g⁰(w)z + 0 > 0,

and this implies v⁺> 0. Hence, any trajectory of solutions will enter the region S.

On region S₃, we have u > u_∗, g(w)− p(u) > 0, v = 0. Thus,

dv⁰ = cv + h(u)[g(w)− p(u)] = 0 + h(u)[g(w) − p(u)] > 0.

Hence, any trajectory of solutions will enter the region S.

In the portion ∂P\ R, we set ∂P = P⁰∪ P¹∪ P¹²∪ P²∪ P³∪ P¹³∪ P¹²³ with P0={(u, v, w, z)|u = 0, w ≥ w^∗, z≥ 0},

P1={(u, v, w, z)|u = u^∗, w > w_∗, z > 0}, P₁₂={(u, v, w, z)|u = u^∗, w = w_∗, z > 0},

P2={(u, v, w, z)|u ∈ (0, u^∗), w = w_∗, z > 0}, P3={(u, v, w, z)|u ∈ (0, u^∗), w≥ w^∗, z = 0} P13={(u, v, w, z)|u = u^∗, w > w∗, z = 0} P123={(u, v, w, z)|u = u^∗, w = w_∗, z = 0}.

Now we investigate the behavior of solutions on each Pi. On region P0, we consider the following three subsets.

(1) Assume v < 0.

Since v < 0, we know that u⁺< 0 and this implies this set belongs to W⁻. (2) Assume v = 0.

In this case, (0, 0, w, z) will stay at W for any (w, z)∈ R². Thus, (0, 0, w, z)∈ J².

(3) Assume v > 0.

Since v > 0, we know that u⁺> 0. (5.61) implies z⁺ > 0 and w⁺> w∗. Thus, the trajectory of solutions will enter the region P .

On region P1, we have g(w)− p(u) > 0 and consider the following three subsets.

(1) Assume v < 0.

Since v < 0, we know that u⁺ < u_∗ and this implies the trajectory of solutions will enter the region P .

(2) Assume v = 0.

(5.63) implies v⁺> 0 and u⁺> u_∗. Thus, the trajectory of solutions will enter the region S.

(3) Assume v > 0.

Since v > 0, we know that u⁺> u_∗. Thus, the trajectory of solutions will enter the region S.

On region P12, we have g(w)− p(u) = 0, w⁺> w∗ and consider the following three subsets.

(1) Assume v < 0.

Since v < 0, we know that u⁺ < u_∗ and this implies the trajectory of solutions will enter the region P .

(2) Assume v = 0.

(5.64) implies v⁺> 0 and u⁺ > u_∗. (5.65) implies [g(w)− p(u)]⁺> 0. Thus, the trajectory of solutions will enter the region S.

(3) Assume v > 0.

Since v > 0, we know that u⁺> u_∗. (5.65) implies [g(w)− p(u)]⁺ > 0. Thus, the trajectory of solutions will enter the region S.

On region P2, we have w⁺> w_∗. Hence, the trajectory of solutions will enter the region P .

On region P3, (5.61) implies z⁺ > 0 and w⁺ > w_∗. Hence, the trajectory of solutions will enter the region P .

On region P13, we have g(w)− p(u) > 0 and consider the following three subsets.

(1) Assume v < 0.

Since v < 0, we know that u⁺< u_∗. (5.62) implies z⁺> 0. the trajectory of solutions will enter the region P .

(2) Assume v = 0.

(3) Assume v > 0.

Since v > 0, we know that u⁺> u∗. Thus, the trajectory of solutions will enter the region S.

In the portion ∂Q\ S, we set ∂Q = Q⁰∪ Q¹∪ Q¹²∪ Q²∪ Q³⁺∪ Q³⁻∪ Q¹³⁺∪ Q¹³⁻∪ P¹²³with Q₀={(u, v, w, z)|u ≤ u^∗, w = 0, z = 0},

Q1={(u, v, w, z)|u = u^∗, w < w_∗, z < 0}, Q₁₂={(u, v, w, z)|u = u^∗, w = w_∗, z < 0}, Q2={(u, v, w, z)|u > u^∗, w = w∗, z < 0}, Q3+={(u, v, w, z)|u > u^∗, 0 < w < w_∗, z = 0} Q₃₋={(u, v, w, z)|u > u^∗, w < 0, z = 0} Q13+={(u, v, w, z)|u = u^∗, 0 < w < w∗, z = 0}.

Q13−={(u, v, w, z)|u = u^∗, w < 0, z = 0}.

Now we investigate the behavior of solutions on each Pi.

On region Q0, The trajectory of solutions are invariant in V and included in J10. On region Q1, we have g(w)− p(u) < 0 and consider the following three subsets.

(1) Assume v > 0.

Since v > 0, we know that u⁺ > u_∗ and this implies the trajectory of solutions will enter the region Q.

(2) Assume v = 0.

(5.63) implies v⁺< 0 and u⁺< u_∗. Thus, the trajectory of solutions will enter the region R.

(3) Assume v < 0.

Since v < 0, we know that u⁺< u_∗. Thus, the trajectory of solutions will enter the region R.

On region Q12, we have g(w)− p(u) = 0, w⁺< w∗and consider the following three subsets.

(1) Assume v > 0.

Since v > 0, we know that u⁺ > u_∗ and this implies the trajectory of solutions will enter the region Q.

(2) Assume v = 0.

(5.64) implies v⁺< 0 and u⁺ < u_∗. (5.65) implies [g(w)− p(u)]⁺< 0. Thus, the trajectory of solutions will enter the region R.

(3) Assume v < 0.

Since v < 0, we know that u⁺< u∗. (5.65) implies [g(w)− p(u)]⁺ < 0. Thus, the trajectory of solutions will enter the region R.

On region Q₂, we have w⁺< w_∗. Hence, the trajectory of solutions will enter the region Q.

On region Q₃₊, (5.61) implies z⁺< 0 and w⁺< w_∗. Hence, the trajectory of solutions will enter the region Q.

On region Q₃₋, (5.61) implies z⁺ > 0 and it does not enter Q immediately. Since u > u_∗, we know that it does not enter P or R immediately. We consider the following four subsets.

(1) Assume v < 0.

Since v > 0, it does not enter S immediately. Thus, the trajectory of solutions does not exit W immediately and it is included in J12.

(2) Assume v≥ 0 and g(w) − p(u) < 0.

Since g(w)− p(u) < 0, it does not enter S immediately. Thus, the trajectory of solutions does not exit W immediately and it is included in J₁₂.

(3) Assume v = 0 and g(w)− p(u) > 0.

(5.63) implies v⁺> 0. Thus, the trajectory of solutions will enter the region S.

(3) Assume v = 0 and g(w)− p(u) = 0.

We have [g(w)− p(u)]⁰= 0 and [g(w)− p(u)]⁰⁰= g⁰(w)z⁰. By (5.61), [g(w)− p(u)]⁺> 0. On the other hand, dv⁰= dv⁰⁰= 0 but dv⁰⁰⁰= h(u)g⁰(w)z⁰> 0. That implies v⁺> 0 and the trajectory of solutions will enter the region S.

On region Q13+, we have g(w)− p(u) < 0 and consider the following three subsets.

(1) Assume v > 0.

Since v > 0, we know that u⁺> u_∗. (5.62) implies z⁺< 0. the trajectory of solutions will enter the region Q.

(2) Assume v = 0.

(5.63) implies v⁺< 0 and u⁺< u_∗. Thus, the trajectory of solutions will enter the region R.

(3) Assume v < 0.

Since v < 0, we know that u⁺< u∗. Thus, the trajectory of solutions will enter the region R.

On region Q13−, we have g(w)− p(u) < 0 and consider the following three subsets.

(1) Assume v > 0.

Since g(w)− p(u) < 0, we know it does not enter S immediately. Since v > 0, we know that u⁺> u∗and it does not enter P or R immediately. (5.62) implies z⁺> 0 and it does not enter Q immediately. Thus, the trajectory of solutions does not exit W immediately and it is included in J₁₁.

(2) Assume v = 0.

(5.63) implies v⁺< 0 and u⁺< u_∗. Thus, the trajectory of solutions will enter the region R.

(3) Assume v < 0.

Since v < 0, we know that u⁺< u_∗. Thus, the trajectory of solutions will enter the region R.

在文檔中 反應擴散方程之進行波解的存在性 (頁 65-73)