 Solution

ID: past-exam-of-the-fixed-point-methods-for-nonlinear-partial-differential-equations-course-of-the-university-of-oxford/2023/3/b/ii/solution

For the spaces in part (b)(i) and fixed

f \in Y

A : X \to X^{*}

⟨ A (u), v ⟩ = \int_{Ω} ∣\nabla u ∣^{2} \nabla u \cdot \nabla v d x + \int_{Ω} u^{7} v d x - \int_{Ω} f v d x .

For each

u

v

, and the estimates in part (b)(i) show

∥ A (u) ∥_{X^{*}} \leq ∥ u ∥_{X}^{3} + C_{8}^{8} ∥ u ∥_{X}^{7} + C_{8} ∥ f ∥_{8/7} .

Hence

A (u)

X^{*}

, and

A

sends bounded sets to bounded sets. The weak formulation is exactly

⟨ A (u), v ⟩ = 0 \forall v \in X, equivalently A (u) = 0 in X^{*} .

The forcing term is included in

A

so that its right side is zero, as requested.

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