Existence of solution of quasilinear transport equation

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Let $a,hinmathcalC^1(mathbbR)$. I found out that a solution of the problem $u_t+a(u)u_x=0$, $u(0,x)=h(x)$ for $xinmathbbR$ can be represented (implicitly) by $$u(t,x)=h(x-ta(u(t,x))).$$
Now, I have to prove that this solution $u$ exists as a function of class $mathcalC^1$ iff $a(h(s))$ is a monotonically increasing function. Does someone have an idea?




pde transport-equation




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  • Hint: Intermediate value theorem
    – Mattos
    Jan 19 '15 at 9:15










  • @Mattos: Thanks, but to which function should it be applied? I did not get the idea - all my attempts failed so far.
    – JohnSmith
    Jan 19 '15 at 14:38











  • A proof can be found here. Another method in related posts uses the implicit functions theorem.
    – Harry49
    Aug 8 at 19:26















up vote
0
down vote

favorite
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Let $a,hinmathcalC^1(mathbbR)$. I found out that a solution of the problem $u_t+a(u)u_x=0$, $u(0,x)=h(x)$ for $xinmathbbR$ can be represented (implicitly) by $$u(t,x)=h(x-ta(u(t,x))).$$
Now, I have to prove that this solution $u$ exists as a function of class $mathcalC^1$ iff $a(h(s))$ is a monotonically increasing function. Does someone have an idea?




pde transport-equation




share|cite|improve this question






















  • Hint: Intermediate value theorem
    – Mattos
    Jan 19 '15 at 9:15










  • @Mattos: Thanks, but to which function should it be applied? I did not get the idea - all my attempts failed so far.
    – JohnSmith
    Jan 19 '15 at 14:38











  • A proof can be found here. Another method in related posts uses the implicit functions theorem.
    – Harry49
    Aug 8 at 19:26













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0
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Let $a,hinmathcalC^1(mathbbR)$. I found out that a solution of the problem $u_t+a(u)u_x=0$, $u(0,x)=h(x)$ for $xinmathbbR$ can be represented (implicitly) by $$u(t,x)=h(x-ta(u(t,x))).$$
Now, I have to prove that this solution $u$ exists as a function of class $mathcalC^1$ iff $a(h(s))$ is a monotonically increasing function. Does someone have an idea?




pde transport-equation




share|cite|improve this question














Let $a,hinmathcalC^1(mathbbR)$. I found out that a solution of the problem $u_t+a(u)u_x=0$, $u(0,x)=h(x)$ for $xinmathbbR$ can be represented (implicitly) by $$u(t,x)=h(x-ta(u(t,x))).$$
Now, I have to prove that this solution $u$ exists as a function of class $mathcalC^1$ iff $a(h(s))$ is a monotonically increasing function. Does someone have an idea?





pde transport-equation





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 8 at 19:34









Harry49

4,7461825




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asked Jan 19 '15 at 9:04









JohnSmith

529312




529312











  • Hint: Intermediate value theorem
    – Mattos
    Jan 19 '15 at 9:15










  • @Mattos: Thanks, but to which function should it be applied? I did not get the idea - all my attempts failed so far.
    – JohnSmith
    Jan 19 '15 at 14:38











  • A proof can be found here. Another method in related posts uses the implicit functions theorem.
    – Harry49
    Aug 8 at 19:26

















  • Hint: Intermediate value theorem
    – Mattos
    Jan 19 '15 at 9:15










  • @Mattos: Thanks, but to which function should it be applied? I did not get the idea - all my attempts failed so far.
    – JohnSmith
    Jan 19 '15 at 14:38











  • A proof can be found here. Another method in related posts uses the implicit functions theorem.
    – Harry49
    Aug 8 at 19:26
















Hint: Intermediate value theorem
– Mattos
Jan 19 '15 at 9:15




Hint: Intermediate value theorem
– Mattos
Jan 19 '15 at 9:15












@Mattos: Thanks, but to which function should it be applied? I did not get the idea - all my attempts failed so far.
– JohnSmith
Jan 19 '15 at 14:38





@Mattos: Thanks, but to which function should it be applied? I did not get the idea - all my attempts failed so far.
– JohnSmith
Jan 19 '15 at 14:38













A proof can be found here. Another method in related posts uses the implicit functions theorem.
– Harry49
Aug 8 at 19:26





A proof can be found here. Another method in related posts uses the implicit functions theorem.
– Harry49
Aug 8 at 19:26
















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