id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
ejde-1105	Xu, Tianyuan; Liu, Gege; Yin, Jingxue	Existence and stability of forced waves for p-Laplace equations in a shifting habitat	2025	16	.pdf	application/pdf	7752	401	81	∫∫ Qτ ( β 2 + u1 + u2 − r(ξ))u2αn(x)e −βtdxdt +D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)e −βtαn(x)dxdt = −D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1 − u2)e −βtα′ n(x)dxdt ≤ D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x) 2|α′ n(x)|e−βtdxdt+ ∫∫ Qτ u2|α′ n(x)|e−βtdxdt. (3.22) Noticing that α′ n(x) = 0 for |x| < n and |x| > n+ 1, we see that D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x) 2|α′ n(x)|e−βtdxdt+ ∫∫ Qτ u2|α′ n(x)|e−βtdxdt ≤ 2D ∫∫ Qτ (|u1x|2(p−1) + |u2x|2(p−1))|α′ n(x)|dxdt+ ∫∫ Qτ u2|α′ n(x)|dxdt ≤ C, where C is independent of n. Letting n → ∞ in (3.22), we obtain∫∫ Qτ u2e−βtdxdt+D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)e −βtdxdt ≤ C. (3.23) Recalling (3.22) and noticing that uix(i = 1, 2) are bounded, we infer that D ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x) 2|α′ n(x)|e−βtdxdt+D ∫∫ Qτ u2|α′ n(x)|e−βtdxdt ≤ D(p− 1)(∥u1x∥L∞ + ∥u2x∥L∞)p−2 ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)|α′ n(x)|e−βtdxdt +D ∫∫ Qτ u2|α′ n(x)|e−βtdtdx ≤ C ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)|α′ n(x)|e−βtdxdt+D ∫∫ Qτ u2|α′ n(x)|e−βtdxdt. From (3.23), the above inequality reduces to C ∫∫ Qτ (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x)|α′ n(x)|e−βtdxdt+D ∫∫ Qτ u2|α′ n(x)|e−βtdxdt → 0, as n → ∞. Now, (3.22) shows that 1 2 ∫ R e−βtu2(x, τ)dx+ ∫∫ Qτ e−βt(u2 + (|u1x|p−2u1x − |u2x|p−2u2x)(u1x − u2x))dxdt ≤ 0, which implies that u1 = u2 in Qτ .	cache/ejde-1105.pdf	txt/ejde-1105.txt