{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

PaperIII_66 (3) - MATHEMATICAL TRIPOS Part III Wednesday 8...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
MATHEMATICAL TRIPOS Part III Wednesday, 8 June, 2011 9:00 am to 11:00 am PAPER 66 REACTION-DIFFUSION EQUATIONS Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Cover sheet None Treasury Tag Script paper You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator. 2 1 On a Hilbert space H consider the abstract non-linear problem d t (0, T ), f : [0, T ) × U → H, (1) dtu(t) = Lu(t) + f (t, u(t)), u(0) = u0 , where U H is an open subset and T > 0. State assumptions on the operator L : D(L) H → H and on f , which are sufficient to prove existence of a unique, classical, local-in-time (i.e. on a time interval [0, t0 ) [0, T ) for t0 small enough) solution for given initial data u0 := D((−L)α ) for α [0, 1). You should give a definition of a classical, local-in-time solution.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}