Complicated dynamics of parabolic equations with simple gradient dependence
In: Transactions of the American Mathematical Society, Jg. 350 (1998), S. 3119-3130
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Zugriff:
Let Q c JR2 be a smooth bounded domain. Given positive integers n, k and ql 0, (X,Y) y U=01 ~1=1 t > O, (X, S) C '9Q. where a(x, y) and al (x, y) are smooth functions. By refining and extending previous results of Polaik we show that arbitrary k-jets of vector fields in Rn can be realized in equations of the form (E). In particular, taking ql _ 1 we see that very complicated (chaotic) behavior is possible for reaction-diffusionconvection equations with linear dependence on Vu. INTRODUCTION Let Q C R2 be a smooth bounded domain. We consider the semilinear parabolic equation ()Ut = UXX + uyy 4a(x, y)u + f (x, y, u, uy), t > O, (x, y) E Q u =0, t > 0, (x, y) E where a: 1R2 --* R is continuous and f: R4 --* R, (x,y,s,w) |-* f(x,y,s,w), is a smooth function (more precisely, f is continuous together with all its partial derivatives with respect to (s, w)). Set X = LP(Q), p > 2, and let A: W2P(Q)nWO'P(Q) -* X be the linear operator defined as -Au = Uxx + uyy + a(x, y)u. Being a sectorial operator, A generates the family X', a > 0, of fractional power spaces. Choosing a with 2 + p < a < 1, we have that X' c C01(Q) with continuous .~~~~~~~~ p inclusion. Define the Nemitski operator f: X' -* X by f (u)(X, y) --f (X, y, 2 (X, y), uy (X, y) Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K20; Secondary 35B40.
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Complicated dynamics of parabolic equations with simple gradient dependence
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Autor/in / Beteiligte Person: | Prizzi, Martino ; Rybakowski, Krzysztof P. |
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Zeitschrift: | Transactions of the American Mathematical Society, Jg. 350 (1998), S. 3119-3130 |
Veröffentlichung: | American Mathematical Society (AMS), 1998 |
Medientyp: | unknown |
ISSN: | 1088-6850 (print) ; 0002-9947 (print) |
DOI: | 10.1090/s0002-9947-98-02294-6 |
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