I new to this stack, and thought my question belongs here.

I am a first-year graduate student currently taking my second course on PDEs (basically converging Evans ch 5 and onwards). I am planning to do research on the analysis of PDEs (No specific topic or advisor yet).

My question is how to really study the subject and understand it. It might seem a dumb question to ask but I am really struggling to understand how do people approach such a course or a subject. Upon glancing at these chapters, I see a lot of function spaces' definitions and long (sometimes tedious) proofs with well-crafted tricks.

More specifically, do I read every proof and try to reproduce it myself later? Do I write down notes while reading (this turned out very time-consuming for me last semester)? Do I try to prove things myself? Do I..

Maybe it would help if some of the experts in PDEs here shared their experiences when they first came in contact with the subject.

Advices are also welcomed as to focusing on PDEs as a research career.

I am new to this stack, and thought my question belongs here.

I am a first-year graduate student currently taking my second course on PDEs (basically covering Evans ch. 5 and onwards). I am planning to do research on the analysis of PDEs (No specific topic or advisor yet).

My question is how to really study the subject and understand it. It might seem a dumb question to ask but I am really struggling to understand how do people approach such a course or a subject. Upon glancing at these chapters, I see a lot of function spaces' definitions and long (sometimes tedious) proofs with well-crafted tricks.

More specifically, do I read every proof and try to reproduce it myself later? Do I write down notes while reading (this turned out very time-consuming for me last semester)? Do I try to prove things myself? Do I...

Maybe it would help if some of the experts in PDEs here shared their experiences when they first came in contact with the subject.

Advice is also welcomed as to focusing on PDEs as a research career.

I new to this stack, and though my question belongs here.

I am a first-year graduate student currently taking my second course on PDEs (basically converging Evans ch 5 and onwards). I am planning to do research on the analysis of PDEs (No specific topic or advisor yet).

My question is how to really study the subject and understand it. It might seem a dumb question to ask but I am really struggling to understand how do people approach such a course or a subject. Upon glancing at these chapters, I see a lot of function spaces' definitions and long (sometimes tedious) proofs with well-crafted tricks.

More specifically, do I read every proof and try to reproduce it myself later? Do I write down notes while reading (this turned out very time-consuming for me last semester)? Do I try to prove things myself? Do I..

Maybe it would help if some of the experts in PDEs here shared their experiences when they first came in contact with the subjects.

Advices are also welcomed as to focusing on PDEs as a research career.

I new to this stack, and thought my question belongs here.

I am a first-year graduate student currently taking my second course on PDEs (basically converging Evans ch 5 and onwards). I am planning to do research on the analysis of PDEs (No specific topic or advisor yet).

My question is how to really study the subject and understand it. It might seem a dumb question to ask but I am really struggling to understand how do people approach such a course or a subject. Upon glancing at these chapters, I see a lot of function spaces' definitions and long (sometimes tedious) proofs with well-crafted tricks.

More specifically, do I read every proof and try to reproduce it myself later? Do I write down notes while reading (this turned out very time-consuming for me last semester)? Do I try to prove things myself? Do I..

Maybe it would help if some of the experts in PDEs here shared their experiences when they first came in contact with the subject.

Advices are also welcomed as to focusing on PDEs as a research career.