Transcribed TextTranscribed Text

- If we look for a radial solution u(x) = v(r) of the nonlinear elliptic equation −∆ = u p in R n , where r = |x| and p > 1, we are led to the nonautonomous ODE (∗) v 00 + n − 1 r v 0 + v p = 0. Show that the Emden-Fowler transformation t := log r, x(t) := e 2t p−1 v(e t ) converts (*) into an autonomous ODE for the new unknown x = x(t). - Find a nonnegative scaling invariant solution having the form u(x, t) = t −α v(xt−β ) for the nonlinear heat equation ut − ∆(u γ ) = 0, where n−2 n < γ < 1. Your solution should go to zero algebraically as |x| → ∞. - Find a solution of −∆u + u n+2 n−2 = 0 in B(0, 1) having the form u = α(1 − |x| 2 ) −β for positive constants α, β. This example shows that a solution of a nonlinear PDE can be finite within a region and yet approach infinity everywhere on its boundary

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:

    for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Differential Equations Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats