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Proof. From the Strong Maximum Principle and the Hopf Boundary Lemma for parabolic equations [7], it follows immediately that u;(x, t) >0 in Q for t>0, i=1, 2. Hence E(u(t)) is well-defined for all t>0. After some tedious calculations, we have It is easy to see that the first integral in the right-hand side of (4.18) is non- positive if

Figure 2 Proof. From the Strong Maximum Principle and the Hopf Boundary Lemma for parabolic equations [7], it follows immediately that u;(x, t) >0 in Q for t>0, i=1, 2. Hence E(u(t)) is well-defined for all t>0. After some tedious calculations, we have It is easy to see that the first integral in the right-hand side of (4.18) is non- positive if

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we obtain the upper bound for any solution of (5.1). then there exist two positive constants C(y) < C(y), which are independent of d, and r,, (i,j =1, 2), such that for any solution u of (3.1) we have
we obtain the upper bound for any solution of (5.1). then there exist two positive constants C(y) < C(y), which are independent of d, and r,, (i,j =1, 2), such that for any solution u of (3.1) we have
Next, we turn our attention to the calculation of index(T,, y*). Recall that y* = G(u*;r,), Le., * is a fixed point of 7,. After some elementary but tedious calculations, we have
Next, we turn our attention to the calculation of index(T,, y*). Recall that y* = G(u*;r,), Le., * is a fixed point of 7,. After some elementary but tedious calculations, we have
Similarly as in the proof of Lemma 5.13, —u<0 is an eigenvalue of I—DT,(w*) if and only if the matrix is singular; that is, |N,(4)|=0 for some 420 and some />0. Hence it suffices to solve the following algebraic equation
Similarly as in the proof of Lemma 5.13, —u<0 is an eigenvalue of I—DT,(w*) if and only if the matrix is singular; that is, |N,(4)|=0 for some 420 and some />0. Hence it suffices to solve the following algebraic equation
It is easy to find that for large ry,
It is easy to find that for large ry,
Then solving |N,(i)| =0 is equivalent to solving the algebraic equation dy = g)(u). In the following, for fixed constants d,, 17,1, 121}, 1o. and large rj, we collect certain properties of g;(4) which will be helpful later in the calcula- tion of index(T,, *). For />1, it is quite complicated to solve (5.26). For this purpose, we set we deduce that for r,, large, the quadratic equation (5.27) has exactly two positive roots, denoted as “Wo, 1, fo, 2. In case when |M| <0, it is easy to see that (5.27) has exactly one positive root. —1 ec UC: ae ye 4 ee qe
Then solving |N,(i)| =0 is equivalent to solving the algebraic equation dy = g)(u). In the following, for fixed constants d,, 17,1, 121}, 1o. and large rj, we collect certain properties of g;(4) which will be helpful later in the calcula- tion of index(T,, *). For />1, it is quite complicated to solve (5.26). For this purpose, we set we deduce that for r,, large, the quadratic equation (5.27) has exactly two positive roots, denoted as “Wo, 1, fo, 2. In case when |M| <0, it is easy to see that (5.27) has exactly one positive root. —1 ec UC: ae ye 4 ee qe
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