Solutions to the Korteweg–de Vries Equation with Initial Profile in $L_1^1 (\mathbb{R}) \cap L_N^1 (\mathbb{R}^ + )$

Cohen, Amy; Kappeler, Thomas (1987). Solutions to the Korteweg–de Vries Equation with Initial Profile in $L_1^1 (\mathbb{R}) \cap L_N^1 (\mathbb{R}^ + )$. SIAM Journal on Mathematical Analysis, 18(4):991-1025.

Abstract

The Cauchy problem for the Korteweg–de Vries equation is considered with initial profile integrable against $(1 + | x |)dx$ on $\mathbb{R}$ and against $(1 + | x |)^N dx$ on $\mathbb{R}^ +$. Classical solutions are constructed for $N \geqq {{11} / 4}$. Under mild additional hypotheses the solution evolves in $L^2 (\mathbb{R})$.

Abstract

The Cauchy problem for the Korteweg–de Vries equation is considered with initial profile integrable against $(1 + | x |)dx$ on $\mathbb{R}$ and against $(1 + | x |)^N dx$ on $\mathbb{R}^ +$. Classical solutions are constructed for $N \geqq {{11} / 4}$. Under mild additional hypotheses the solution evolves in $L^2 (\mathbb{R})$.

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